By Approriate scalling in Yamabe equation, I have a question: Let $latex {M}&fg=000000$ be a homogeneous Lie group of step 2 (for example, Heisenberg group). How to find the constant of the conformal laplacian in this case?

## Minlin-Hormander theorem for Morrey spaces

### Featured

In the last post we given a proof of the following theorem.

Theorem 1 (Minlin-Hormander theorem)Let and . If for we have

then is a Fourier multiplier on , . In particular, is a Fourier multiplier on if , that is, if

In this post, we would like to extend this theorem for a class of Morrey-type spaces. In other words

Theorem 2Let and . If , the Fourier multiplier on can be extended to , that is, there exists such that

**Proof.** A proof can be found in [Taylor] (see also [Kozono1, Kozono2]. For reader convenience we give some details. This key theorem was important in my recent article in join with Ferreira, Lucas C.F.

We have that is a convolution operator with kernel . As it follows from [p.26, Stein] that

Based in (4) and Minlin-Hormander theorem we obtain a proof of the Theorem 2. Indeed, firstly one splits as

where

Defining and easily gets by Hölder inequality and estimate (4) the following

where denotes the conjugate exponent of . Using Tonelli’s theorem and once more estimate (4) we have

Now we ready to proof the theorem. By Minlin-Hormander theorem and the last estimate one has

which yields (3), because the series above is convergent.

## Minklin-Hormander

### Featured

**1. Minklin-Hormander type symbols**

Let be vector-spaces of measurable functions from to itself and let be a bounded linear operator from to . Recall that is called translation invariant if for all and . Let and with , we found that each such operator is determined by a certain tempered distribution such that for every (Schwartz space). So taking Fourier transform into we have . This motivate us to define a *Fourier multiplier* as a map given by

where is a tempered distribution and denotes the Fourier transform . We refer to as *symbol* of , sometimes one writes as to relate it with more general operators so-called pseudo-differential operators. No standard example of such *symbols* is, for ,

In the limit case, , the above symbol can be written as , where denotes the characteristic function of unit disk . It’s well-known that the condition and is necessary for be a *Fourier multiplier* on (see e.g., [1]). However, Fefferman (see [2],[3]) gave an intricate proof which show us that this condition is not sufficient, that is, he showed that the operator does not extend to a bounded operator on for any and . This result give us a negative answer to the famous disk conjecture which states that is bounded on for . In this post we will work with symbols more regular than (1) such as Minklin symbols .

In a few months ago, based on Littlewood-Paley theorem, we gave a proof that the operator is a *Fourier multiplier* from to itself (see Theorem 7) provided that and satisfies

for and . In this post will be showed that if , then its satisfies the inequality (2). Hence, by Theorem 7 one has the following classical Minklin-Hormander theorem.

Theorem 1 (Minklin-Hormander theorem)Let and away from the origin. If for we have

then is a Fourier multiplier on , . In particular, is a Fourier multiplier on if , that is,

Let us recall some important definitions. Let and , a function lies in *Sobolev spaces* if for every with there exists such that

Here, we use the standard notations

and we say that is the derivative of in distribution sense (more precisely in ) and we write in to mean (5). The space equipped with the norm

is a Banach space. Also, notice that *Sobolev spaces* coincide with inhomogeneous *fractional Sobolev spaces* because of the norm equivalence

It follows that

Using Leibniz’s formula we written the term as

By observing that on and zero otherwise, we get

Now making the change of variable one has . Hence, by (3) it follows that

Let . Therefore, inserting (7) into (6) easily gets

and Theorem 1 is a consequence of the Theorem 7 as we desired. Notice that if ,

Hence which implies (8).

## Heat-wave equation in Morrey spaces

### Featured

**1. A Heat-Wave equation in Morrey spaces **

This notes is an exposition of the article written by me in join with full professor Lucas Catão de Freitas Ferreira and therefore no proofs will be given in general.

Throughout this notes, the symbol stands for the set of all natural numbers. For , we always let be -dimensional Euclidean space with Euclidean norm denoted by and endowed with Lebesgue measure . Also stands for -dimensional upper half-space with Lebesgue measure . Let be a vector field on upper half-space , we denote . Here is so-called absolute value of .

Here we are interested in a semilinear integro-partial differential equation, which interpolates the semilinear heat and wave equations which reads as follows

where denotes the Gamma function, is the Laplacian in the -variable and is a fluid at time and position that assumes the given data (initial velocity) . This equation is formally equivalent to the initial value problem for the time-fractional partial differential equation (FPDE)

where and stands for the Riemann-Liouville derivative of order given at (16).

Differential equations of time-fractional order appear naturally in several fields such as physics, chemistry and engineering by modelling phenomena in viscoelasticity, thermoelasticity, processes in media with fractal geometry, heat flow in material with memory and many others. The two most common types of fractional derivatives acting on time variable are those of Riemann-Liouville and Caputo. We refer the surveys Kilbas and Povstenko in which the reader can find a good bibliography for applications on those fields. Models with fractional derivatives can naturally connect structurally different groups of PDEs and their mathematical analysis may give information about the transition (or loss) of basic properties from one to another. Two groups are the parabolic and hyperbolic PDEs whose well-posedness and asymptotic behavior theory presents a lot of differences. For instance, in weak- Besov-spaces, Morrey spaces and other ones, there is an extensive bibliography for global well-posedness and asymptotic behavior for semilinear heat equations (and other parabolic equations). On the other hand, for semilinear wave equations, although there exist results in , weak and Besov spaces, there is no results in Morrey spaces. The main reason is the loss of decay of the semigroup (and its time-derivative) associated to the free wave equation, namely and So, it is natural to wonder what would be the behavior of the semilinear (FPDE) in the framework of Morrey spaces, which presents a mixed parabolic-hyperbolic structure.

Between interesting points obtained, let us comment about technical ones. Due to the semigroup property (21), further restrictions appear in our main theorems in comparison with classical nonlinear heat equations. Making the derivative index go from to the estimates and corresponding restrictions become worse, and they are completely lost when reaches the endpoint (see Lemmas 4, 5). The proof of the pointwise estimate (23) shows that the *worst parcel* in (21) is the term (see (21)). Then In particular, notice that for (heat semigroup) the upper bound on parameter is not necessary, that is, one can take Finally, based on above observations, our results and estimates suggest the following: *the semilinear wave equation () in is not well-posed in Morrey spaces*. The mathematical verification of this assertion seems to be an interesting open problem.

**2. Morrey spaces **

In this section some basic properties about Morrey spaces are reviewed. For further details on theses spaces, the reader is referred to Kato, Peetre, Taylor. Let denote the open ball in centered in the origin and with radius . For two parameters and , we define the Morrey spaces as the set of functions such that

where denotes a constant independent of and . The space endowed with the norm

is a Banach space. For and the homogeneous Sobolev-Morrey space is Banach with norm

Taking , stands for the total variation of on and is a space of signed measures In particular, when is the space of finite measures. For and is the homogeneous Sobolev space With the natural adaptation in (3) for the space corresponds to .

Morrey spaces present the following scaling

and

where the exponent is called scaling index. We have that

Let us define the closed subspace of (denoted by ) by means of the property if and only if

This subspace is useful to deal with semigroup of convolution operators when the respective kernels present a suitable polynomial decay at infinity. In general, such semigroups are only weakly continuous at in but they are -semigroups in as it is the case of . This property is important in order to derive local-in-time well-posedness for PDEs.

Morrey spaces contain Lebesgue and Marcinkiewicz spaces with the same scaling indexes. Precisely, we have the continuous proper inclusions

where and (see e.g. Miyakawa or MF de Almeida and LFC Ferreira).

In the next lemma, we remember some important inequalities and inclusions in Morrey spaces, see e.g. Kato, Taylor.

Lemma 1Suppose that , and .

- (Inclusion) is decreasing in , i.e., if and then
- (Sobolev type embedding) If , and then

- (Holder inequality) If and then and

- (Homogeneous function) Let , and . Then , for all .

We finish this section by recalling estimates for certain multiplier operators on see e.g. Kozono-Yamazaki, Kozono-Yamazaki, Taylor for lemma below.

In the next posts, inspired in the work of Taylor, we given a proof based in the **Theorem 7** given in the post A fractional Hörmander type multiplier theorem.

Lemma 2Let and and Assume that there is such thatfor all with and for all Then the multiplier operator on is bounded from to and the following estimate hold true

where is the set of equivalence classes in modulo polynomials with variables.

**3. Mittag-Leffler Function**

Differential equations of time-fractional order is old, but even being old, the fractional calculus is little studied by mathematicians. Possibly, because many of them are unfamiliar with this topic and its applications in various sciences. After Liouville and Riemann, deep developments was obtained by many authors. Today, time-fractional order integral and derivative is known as Riemann-Liouville’s integral and derivative. More precisely, let be a Lebesgue integrable function in and , Riemann-Liouville’s integral is defined by

and Riemann-Liouville’s derivative by

where , , being denoted by so-called Gamma function.

Hardy and Littlewood give wider properties about this integrals. His showed, for instance, the boundedness of from to , for . This result, is well known as Hardy-Littlewood-Sobolev theorem, Sobolev because of its importance in the theory of fractional Sobolev Spaces.

Related the theory of partial differential equations, much works is devoted to find an unified theory of Green functions associated to fractional problems. In this way, fractional diffusion-wave equation is a celebrity, and an unified theory of the heat equation and wave equation was obtained. Fujita, Schneider and Wyss, for instance, founds the special function to represent the Green function associated to the linear part of the diffusion-wave problem (FPDE)

such special function, is knew as Mittag-Leffler function. More precisely,

and the Green function, defined via Fourier inversion, is given by

which generate the following semigroup of operators ,

In what follows, we recall some functions which is useful to handle the symbol of the semigroup (see (19)). For let us set

and

Lemma 3Let and be as in (18). We have thatand

for all Moreover, , is a probability measure, and

*Proof:* Except for (22), all properties contained on the statement can be found in Fujita and Hirata-Miao when and respectively.

In order to prove (22), we use Fourier inversion and (18) to obtain

The desired identity follows by taking in the last equality.

**4. Some estimates**

The aim of this section is to derive estimates for the semigroup For that matter we will need pointwise estimates for the fundamental solution in Fourier variables.

Lemma 4Let and There is such thatfor all with and for all

In the sequel we prove key estimates on Morrey spaces for the semigroup

Lemma 5Let , , and . There exists such thatfor all

*Proof:* Let and defined through Consider the multiplier operators

that is,

where the symbol of is . Lemma 4 implies that satisfies (13) with Then, we use (5) to obtain

and therefore,

Now, using Sobelev embedding (11), and afterwards (25), we obtain

because of (26).

**5. Self-similarity and symmetries for a fractional-wave equation**

The problem (1) can be formally converted to the integral equation (see [Hirata-Miao])

with

which should be understood in the Bochner sense in Morrey spaces, being . Throughout this notes a mild solution for (1) is a function satisfying (27). We shall employ the Kato-Fujita method (see [Kato]) to integral equation to get our results.

From now on, we perform a scaling analysis in order to choose the correct indexes for Kato-Fujita spaces. A simple computation by using

shows that the indexes and are the unique ones such that the function given by

is a solution of that, for each whenever is also. The scaling map for (29) is defined by

Making in (31) one obtains the following scaling for the initial condition

Solutions invariant by (31), that is

are called self-similar ones. Since we are interested in such solutions, it is suitable to consider critical spaces for and , i.e., the ones whose norms are invariant by (31) and (32), respectively.

Consider the parameters

and let stands for the class of bounded and continuous functions from to a Banach space We take belonging to the critical space and study (27) in the Kato-Fujita type space

which is Banach with the norm

Notice that the norm (36) is invariant by scaling transformation (31).

From Lemma 1, a typical data belonging to is the homogeneous function

where and is a bounded function on sphere We refer the book [Gigabook, chapter 3] for more details about self-similar solutions and PDE’s.

Our well-posedness result reads as follows.

Theorem 6 (Well-posedness)Let , , , and . Suppose that and

- (i) (Existence and uniqueness) There exist and such that if then the equation (1) has a mild solution , which is the unique one in the ball . Moreover, in as
- (ii) (Continuous dependence on data) Let The data-solution map is Lipschitz continuous from to .

Remark 1

- (i) With a slight adaptation of the proof of Theorem 6, we could treat more general nonlinearities. Precisely, one could consider (1) and (29) with instead of , where , and there is such that
- (ii) (Local-in-time well-posedness) A local version of Theorem 6 holds true by replacing the smallness condition on initial data by a smallness one on existence time . Here we should consider the local-in-time spaceand such that In particular, this condition is verified when belongs to the closed subspace (see (8)).

Let be the orthogonal matrix group in and let be a subset of A function is said symmetric and antisymmetric under the action of when and , respectively, for every .

Theorem 7Under the hypotheses of Theorem 6.

- (i) (Self-similarity) If is a homogeneous function of degree , then the mild solution given in Theorem 6 is self-similar.
- (ii) (Symmetry and antisymmetry) The solution is antisymmetric (resp. symmetric) for , when is antisymmetric (resp. symmetric) under
- (iii) (Positivity) If and (resp. ) then is positive (resp. negative).

Remark 2(Special examples of symmetry and antisymmetry)

- (i)
The casecorresponds to radial symmetry. Therefore, it follows from Theorem 7 (ii) that ifis radially symmetric thenis radially symmetric for.- (ii) Let be the reflection over the origin and let be the identity map. The case corresponds to parity of functions, that is, is even and odd when and , respectively. So, from Theorem 7 (ii), we have that
the solutionis even (resp. odd) forwhenis even (resp. odd).

## A fractional Hörmander type multiplier theorem

### Featured

**— 1. A fractional Hörmander type multiplier theorem —**

In a few months ago we deal with translation invariant operators in , see previous post. More precisely, was shown that there exists a tempered distribution such that . However, what does happens with to get -boundedness of ? Such questions belongs to a fruitiful area of Fourier analysis which many personages work in that, such as Marcinkiewicz, Minhklin, Hörmander, Lizorkin and more recentily Fefferman’s work about ball multiplier conjecture. In this post will be initiated the saga related to spaces so far. Before, we recall some important definitions. Let , we set

for any . Also, recall that

Then via Theorem 3 the operator may be written in as

where . In fact, by observing that

we have from (2) that

being the inverse Fourier transform of . We call the operator in (3) of **multiplier** operator associated to symbol and we will written it as . Also, we say that is a **multiplier** on , if for all the map is bounded on .

Example 1The Fourier transform in classical sense ofis not possible, because not belong to any space. However, Fourier transform of in distribution sense is given by

for more details, see notes of Iannis Parissis.

Let , the Bessel space is the space of functions such that which endowed with norm

is a Banach space. The following proposition give a condition on symbol for which is a **multiplier** on .

Proposition 1Let and . Then is a multiplier on , for .

*Proof:* In view of , where . From Young’s inequality, just we need to show . To do this, let . As , from Cauchy-Schwartz inequality one gets

Example 2Let be bounded, that is, . We claim thatIt’s quite easy to get via Plancherel’s theorem. It remains to get the other inequality. Let be a ball in with radius . Let such that and on . Note that

The Lebesgue differentiation’s theorem yields

in other words, .

Let be the set of all tempered distributions away from origin of satisfying the estimate

for all multi-index with . In the next sections we give a much weaker hypotheses on symbol , via Littlewood-Paley theory, which imply such as so-called symbol-H\”{o}rmader condition and . In all that follows, the class denotes the set of **multipliers** operators with symbol belonging to .

**— 1.1. Littlewood-Paley theory —**

Let be radial Schwartz function supported in which is identically on . To get that, recall of the topological Urysohn’s lemma. For all , set

Notice that , because and for, respectively, and . Moreover, the family of functions forms a *partition of unity*

Next, for we define the cut-off operator or Littlewood-Palay operator as

where . Notice that is supported in . Indeed, by Fourier property

it follows that . Therefore, if is supported in the annulus then the inverse Fourier transform is supported in the annulus

In other words, the operator isolate the part of a function being each one concetrated near annulus .

Proposition 2The cut-off operator is a self-adjoint operator,for any and .

*Proof:* The prove is a consequence of the radial property of . For that, firstly, shall be observed that

when we have in mind (1). Now, radial property of and (4) implies that

The decomposition of a tempered distribution into a sum of “long pieces” supported in the annulus is known as Littlewood-Paley decomposition, because of the famous works de Littlewood and Paley related to Fourier and Power series, see xx. More precisely, Littlewood-Paley decomposition of a tempered distribution is defined as follows

However, there is tempered distributions for which Littlewood-Paley decomposition fail on . For instance take the distribution and note by from Proposition 2 and that

that is, , . To overcome this discomfort, we introduce in the equivalence class given by

where denotes the set of polynomials ,

with complex coefficients and a non-negative integer. The space endowed with will be denoted by .

Proposition 3Let be the space of Schwartz functions such thatThen is a subspace of which has

in sense of isomophism.

*Proof:* The prove is standard, consider the identification map which goes an element into the equivalence class that contains it. The kernel of the map is .

The aim is that, for any we have

This follows by observing that

where is that radial Schwartz function suported into a compact set and equal to in . To show (7), recall that a distribution is a tempered distribution if

Easily, via identification (6), this indue in nature’s way the convergence in . Thus, it is quite easy to show . It remains to show

Let us fixe such that and write

because for we have and on ball . Thus, given one has and the its Littlewood-Paley decomposition is null in , because of

and

The last equality is obtained via

where is the Dirac mass and don’t forget that , which is consequence of properties

- (i) for every multi-index
- (ii) for every multi-index

for all .

Proposition 4If , then agrees with function away from the origin and satisfies the following pointwise estimatefor all multi-index , where .

*Proof:* Let , then for all this give us

where each peace is supported on annulus . So makes sense to define,

If , for one has

It follows that

If we put into the last equality above, for all multi-indices and non-negative we have

We split into two sums as

For and into (10), respectively, we get

and

This inequalities says us that converges absolutely and uniforms in . It follows that converges in to a function which also satisfies the estimate

As converges to , then and we obtain the desired estimate.

In what follows, denotes the space of Bochner integrable functions from to a Banach space and we write

Theorem 5 (Littlewood-Paley Theorem)Let the Littlewood-Paley operator. If , there exists a positive constant such thatMoreover,

*Proof:* The inequality (13) follows from Theorem 5.4 in [1]. In fact,

in other words, the map is bounded from to . It remains to show that the kernel of the map satisfies the H\”{o}rmander condition, which can be obtained from estimate

Indeed, by mean value theorem one has

The inequalities above show us that the kernel is a singular kernel from to and the inequality (13) follows from Theorem 5.4 in [1].

From now on, will be showed the estimate (14). To this end, we remember of which give us

Recall that , where is a Schwartz function. Therefore, for all multi-index , that is, . And by Proposition 4 follows that there exists a positive constant for which

Setting

and recalling that , we have via inequalities (15) and (16), respectively, that

Let be in , where . For each , let and . From self-adjointness of , follows that

By duality of and Cauchy-Schwartz inequality, respectively, one has

where we use the Hölder inequality and first Littlewood-Paley inequality (13) above. If we replace by in (16) such that and we put , it follows that

as desired. It remains to get . To do this, let

and set

Then on annulus , that is, . Indeed, since and since . This finish our prove.

Lemma 6Let and . Set . If , then there exists a positive constant (independent of , and ) such thatwhere denotes the Hardy-Littlewood maximal function,

*Proof:* Let , then the kernel of satisfy . Therefore, by Hölder inequality and Plancharel theorem we obtain

where and . It follows by Fubini theorem that

The last inequality is obtaind from splitting into lattices of length , see figure below.

Indeed,

where is a integer such that . Setting it leads us to . By definition of and (21), it follows that

the last series converges, because of .

The next theorem is a fractional variant of Hörmander multiplier theorem.

Theorem 7Let be such thatfor and let . Then the operator is a multiplier on .

*Proof:* From Littlewood-Paley inequality (14),

Thus,

where and is a Littlewood-Paley operator such that given in paragraphs above. Now, we write

If , by Riesz representation theorem there exists satisfying and for which

As the symbol of is and , for all . The Lemma 6 gives

As , by Hardy-Littlewood theorem and Hölder’s inequality we get

because the Littlewood-Paley inequality (13) still hold with in place of . Inserting the last inequality into (23) we have

for all . The case follows by standard duality argument and shall be remarked only that the symbol of form adjuint operator is and hence is bounded by arguments above. The case it’s a small exercise to reader.

In the next post shall be showed that fractional condition (22) implies so-called symbol Hörmander condition in Theorem 2.5 and also Minhklin condition for . Some small applications to PDEs equations will be mentioned finally.

# Subcriticality and supercriticality

Originally posted on lim Practice= Perfect:

Consider the equation

$latex displaystyle Delta u=u^ptext{ on }mathbb{R}^n&fg=000000$

usually we call the equation is subcritical when $latex {p<frac{n+2}{n-2}}&fg=000000$, supercritical when $latex {p>frac{n+2}{n-2}}&fg=000000$. The reason comes from the scalling the solution. Suppose $latex {u(x)}&fg=000000$ is a solution of the equation, then $latex {u^lambda(x)=lambda^{frac{2}{p-1}}u(lambda x)}&fg=000000$ is another solution. Consider the energy possessed by $latex {u}&fg=000000$ around any point $latex {x_0}&fg=000000$ of radius $latex {{lambda}}&fg=000000$ can be bounded

$latex displaystyle int_{B_{lambda}(x_0)}|nabla u(x)|^2dxleq E&fg=000000$

when $latex {lambdarightarrow 0}&fg=000000$, we scale $latex {B_lambda(x_0)}&fg=000000$ to $latex {B_1(x_0)}&fg=000000$, then $latex {u}&fg=000000$ will become $latex {u^lambda}&fg=000000$ in order to be a solution and $latex {u^lambda}&fg=000000$ lives on $latex {B_1(x_0)}&fg=000000$. While the energy will be

$latex displaystyle int_{B_{1}(x_0)}|nabla u^lambda(x)|^2dx=lambda^{frac{4}{p-1}+2-n}int_{B_{lambda}(x_0)}|nabla u(y)|^2dy&fg=000000$

If the $latex {delta=frac{4}{p-1}+2-n<0}&fg=000000$, which is $latex {p> frac{n+2}{n-2}}&fg=000000$, the energy bound of $latex {u^lambda}&fg=000000$ will become $latex {lambda^delta E}&fg=000000$. Since $latex {lambdarightarrow 0}&fg=000000$, the bound deteriorates by ‘zooming in’. In this case, we call the equation is…

View original 35 more words

# Smoothness for small initial data

Originally posted on lim Practice= Perfect:

$latex displaystyle begin{cases}u_t+ucdotnabla u-nuDelta u+nabla p=0div u=0u(x,0)=u_0end{cases}text{ on }mathbb{R}^3&fg=000000$

**Thm(Kato):** If $latex {u_0in dot{H}^{1/2}}&fg=000000$ is small, then the solution with initial data $latex {u_0}&fg=000000$ is global smooth.

*Proof:* $latex {Lambda u=sqrt{-Delta}u}&fg=000000$ is a well defined operator on $latex {dot{H}^{1/2}}&fg=000000$. Multiplying $latex {Lambda u}&fg=000000$ on both sides, we get

$latex displaystyle frac{1}{2}frac{d}{dt}||u||_{dot{H}^{1/2}}^2+||u||_{dot{H}^{3/2}}^2leq int |u||nabla u||Lambda u|leq ||u||_{L^{3}}||nabla u||_{L^3}||Lambda u||_{L^3}&fg=000000$

By the Sobolev embedding,

$latex displaystyle ||u||_{L^3}leq C||u||_{dot{H}^{1/2}}&fg=000000$

So we can get

$latex displaystyle ||u||_{L^{3}}||nabla u||_{L^3}||Lambda u||_{L^3}leq C||u||_{dot{H}^{1/2}}||u||_{dot{H}^{3/2}}^2&fg=000000$

Let $latex {G=||u||_{dot{H}^{3/2}}^2}&fg=000000$, $latex {N=||u||_{dot{H}^{1/2}}^2}&fg=000000$,

$latex displaystyle frac{1}{2}frac{d}{dt}N+G(1-Csqrt{N})leq 0&fg=000000$

If $latex {N(0)<1/C^2}&fg=000000$ is small enough, then $latex {N'(t)}&fg=000000$ will be negative thus $latex {N(t)<1/C^2}&fg=000000$ and small enough. Consequently, $latex {||u||_{L^3}leq C||u||_{dot{H}^{1/2}}}&fg=000000$ will also be small. While from the NS equation,

$latex displaystyle frac{1}{2}frac{d}{dt}|nabla u|_{L^2}^2+|Delta u|_{L^2}^2leqint|u||nabla u||Delta u|leq C||u||_{L^3}||Delta u||_{L^2}^2&fg=000000$

When $latex {||u||_{L^3}}&fg=000000$ small enough, the right hand side can be absorded by left hand side. Thus $latex {|nabla u|_{L^2}^2}&fg=000000$ will be bounded. And from…

View original 22 more words

# A forum on mathematical publishing

“Our hope is that this might provide a better home for more focused discussion, and a place for people who want to coordinate concrete next steps in reforming mathematical publishing. Come in and join us!”

Originally posted on Secret Blogging Seminar:

There’s been lots of great discussion on the future of mathematical publishing in recent weeks, largely inspired by the boycott of Elsevier (1) (2) (3). Mostly this has been happening on blogs, particularly Tim Gower’s, but also here and a number of other places. There’s a nice index of this discussion in a wiki page on Michael Nielsen’s site, to the extent that it’s possible to index a discussion happening all over the internet!

I think a lot of people find it somewhat frustrating that this discussion is predominantly happening in blog comment threads, however. It’s hard to maintain conversations, and almost impossible to coordinate people with similar interests and concerns. Andrew Stacey and I thought that it might be helpful to set up a forum (like the nForum, associated the to nCafe, or meta.mathoverflow.net) to alleviate this.

Thus, please check out

View original 43 more words

# Lorentz spaces basics & interpolation

a great post, in the future I will post some important estimates for semilinear evolution equations on euclidian spaces and half-euclidian spaces, such as, Heat estimates, wave estimates, Yamazaki estimates, etc.

Originally posted on Almost Originality:

(Updated with endpoint $latex {q = infty}&fg=000000$)

I’ve written down an almost self contained exposition of basic properties of Lorentz spaces. I’ve found the sources on the subject to leave something to be desired, and I grew a bit confused at the beginning. Therefore this relatively short note (I might be ruining someone’s assignments out there, but I think the pros of writing down everything in one place balance the cons).

Here’s a link to the pdf version of this post: Lorentz spaces primer

**1. Lorentz spaces **

In the following take $latex {1< p,q < infty}&fg=000000$ otherwise specified, and $latex {(X, |cdot|)}&fg=000000$ a $latex {sigma}&fg=000000$-finite measure space with no atoms.

The usual definition of Lorentz space is as follows:

Definition 1The space $latex {L^{p,q}(X)}&fg=000000$ is the space of measurable functions $latex {f}&fg=000000$ such that$latex displaystyle |f|_{L^{p,q}(X)}:= left(int_{0}^{infty}{t^{q/p}{ f^ast (t)}^q},frac{dt}{t}right)^{1/q} < infty,&fg=000000$

where $latex {f^ast}&fg=000000$ is the decreasing rearrangement…

View original 3,176 more words

# Measure theory is a must

Originally posted on Noncommutative Analysis:

###### [This post started out as an introduction to a post I was planning to write on convergence theorems for the Riemann integral. The introduction kind of got out hand, so I decided to post it separately. Since I have to get back to my real work, I will postpone writing that post on convergence theorems for the Riemann integral for another time, probably during the Passover break (but in any case before we need them for the course I am teaching this term, Calculus 2)].

Mathematicians love to argue about subjective opinions. One of the most tiresome and depressing subjects of debate is “What should an undergraduate math major curriculum contain?”

View original 1,019 more words

# Elsevier journals — some facts

Originally posted on Gowers's Weblog:

A little over two years ago, the Cost of Knowledge boycott of Elsevier journals began. Initially, it seemed to be highly successful, with the number of signatories rapidly reaching 10,000 and including some very high-profile researchers, and Elsevier making a number of concessions, such as dropping support for the Research Works Act and making papers over four years old from several mathematics journals freely available online. It has also contributed to an increased awareness of the issues related to high journal prices and the locking up of articles behind paywalls.

However, it is possible to take a more pessimistic view. There were rumblings from the editorial boards of some Elsevier journals, but in the end, while a few individual members of those boards resigned, no board took the more radical step of resigning en masse and setting up with a different publisher under a new name (as some journals have…

View original 10,728 more words