XVIII EARCO 2018

Rogers-Shephard inequality is a classical inequality in convex geometry which provides, for any convex set $K\subseteq\mathbb{R}^n$, an upper bound for the volume of the difference body $K-K$ in terms of the volume of the convex body $K$. More precisely, one has that \[ |K-K|\leq {2n\choose n}|K|, \] with equality if and only if $K$ is a simplex. Rogers and Shephard proved this inequality in 1958 and soon after they proved an inequality which relates the volume of $K$ with the volume of a projection onto a $k$-dimensional subspace and an orthogonal section, obtaining as a consequence several geometric inequalities including the aforementioned one. In this talk we will study Rogers-Shephard inequalities in the more general context of log-concave functions, obtaining in addition new geometric inequalities relating the volume of $K$ with the volumes of sections and projections onto subspaces which are not necessarily orthogonal.

Given a basis $\mathcal{B}=(\mathbf{x}_n)_{n=1}^\infty$ of a Banach space $\mathbb{X}$, for each $m\in\mathbb{N}$ the $m$th-conditionality constant $k_m[\mathcal{B},\mathbb{X}]$ is the smallest constant $C$ for which the inequality \[ \left\Vert \sum_{n\in A} a_n \, \mathbf{x}_n\right\Vert \le C \left\Vert f \right\Vert \] holds for every $f=\sum_{n=1}^\infty a_n \, \mathbf{x}_n\in\mathbb{X}$ and every $A\subseteq\mathbb{N}$ of cardinality at most $m$. Any basis $\mathcal{B}$ of any Banach space $\mathbb{X}$ fulfills the inequality $k_m[\mathcal{B},\mathbb{X}]\lesssim m$ for $m\in\mathbb{N}$. However, the conditionality constants of quasi-greedy bases, that is, bases for which the thresholding greedy algorithm converges, verify a significantly better estimate. This estimate can be improved when the Banach space is super-reflexive. In this talk we present a survey of the main recent advances of this subject, situated halfway between approximation theory and Banach space theory.

Fractional integrals and the classical fractional maximal function are smoothing operators, in the sense that they map Lebesgue spaces into first order Sobolev spaces. We show that this phenomenon continues to hold for the fractional spherical maximal function when the dimension of the ambient space is greater than or equal to 5. A key element in the proof is a local smoothing estimate for the wave equation. This is joint work with Joao P. G. Ramos and Olli Saari.

Hedenmalm, Lindqvist and Seip introduced in 1997 the Hardy space of Dirichlet series $\mathcal{H}^2$ defined as the set of Dirichlet series \[ f(s)=\sum_{n=1}^\infty a_nn^{-s} \] whose coefficients belong to $\ell^2$. On the other hand, the Cesàro sequence space $ces_p$, for $1 < p < \infty$, consisting of all complex sequences $a=(a_n)_{n=1}^\infty$ such that \[ \|a\|_{ces_p}=\left(\sum_{n=1}^\infty\bigg(\frac1n\sum_{k=1}^n|a_k|\bigg)^p \right)^{1/p} < \infty, \] is a Banach lattice that arises naturally from Hardy's result on the boundedness on $\ell^p$ of the Cesàro operator.

We consider the space $\mathcal{H}(ces_p)$ formed by all Dirichlet series whose coefficients belong to $ces_p$, and identify its multiplier algebra \[ \mathcal{M}(\mathcal{H}(ces_p))=\{ g: fg\in\mathcal{H}(ces_p) \mbox{ for every } f\in\mathcal{H}(ces_p)\}. \]

This work is part of a Ph.D. Thesis of the author which is being prepared under the supervision of Guillermo Curbera and Olvido Delgado.

Calderón [Ca] studied interpolation properties of compact bilinear (or multilinear) operators in his foundational paper on the complex interpolation method. The case of the real method was investigated more recently by Fernandez and Silva [FS] and Fernández-Cabrera and Martínez [FM, FM2]. An important motivation for this research has been the fact shown in the last few years that compact bilinear operators occurs rather naturally in harmonic analysis. See, for example, the paper by Bényi and Torres [BT] where they show that commutators $S$ of Calderón-Zygmund bilinear operators are compact when acting from $L_p\times L_q$ into $L_r$ provided that $1\leq r < \infty, 1 < p,q < \infty$ and $1/p + 1/q = 1/r$.

In this talk we will focus on the results of the joint paper with Fernández-Cabrera and Martínez [CFM] where we study the interpolation properties of compact bilinear operators by the real method among quasi-Banach couples. As an application of the results we show that commutators of Calderón-Zygmund bilinear operators $S:L_p \times L_q \to L_r$ are also compact provided that $1/2 < r < 1, 1 < p,q < \infty$ and $1/p + 1/q = 1/r$.

Referencias.

[BT] Á. Bényi, R.H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3609-3621, DOI 10.1090/S0002-9939-2013-11689-8. MR3080183

[Ca] A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math.24 (1964), 113-190, DOI 10.4064/sm-24-2-113-190. MR0167830

[CFM] F. Cobos, L.M. Fernández-Cabrera, A. Martínez, Interpolation of compact bilinear operators among quasi-Banach spaces and applications, preprint.

[FS] D.L. Fernandez, E.B. da Silva, Interpolation of bilinear operators and compactness, Nonlinear Anal. 73, (2010), no.2, 526-537, DOI 10.1016/j.na.2010.03.049. MR2650835

[FM] L.M. Fernández-Cabrera, A. Martínez, On interpolation properties of compact bilinear operators, Math. Nachr. 290 (2017), no. 11-12, 1663-1677, DOI 10.1002/mana.201600203. MR3683453

[FM2], L.M. Fernández-Cabrera, A. Martínez, Real interpolation of compact bilinear operators, J. Fourier Anal. Appl. (2017), 1-23, DOI 10.1007/s00041-017-9561-7.

Our setup is a natural class of operators $T$, which we think of as general (not necessarily of tensor product or convolution type) bilinear bi-parameter singular integrals on the product space $\mathbb R^{n+m} = \mathbb R^n \times \mathbb R^m$. Starting from $T1$ type assumptions, we show a representation theorem for these operators using dyadic model operators. For singular integrals that are free of full paraproducts we use the representation to show weighted estimates $L^p(w_1) \times L^q(w_2) \to L^r(v_3)$, where $1 < p, q < \infty$, $1/2 < r < \infty$, $1/p+1/q = 1/r$, $w_1 \in A_p(\mathbb R^n \times \mathbb R^m)$, $w_2 \in A_q(\mathbb R^n \times \mathbb R^m)$ and $v_3 := w_1^{r/p} w_2^{r/q}$. Analogous unweighted estimates are obtained for all singular integrals. We also consider mixed-norm estimates. As an application we give a new proof of a result of Muscalu-Pipher-Tao-Thiele: a bi-parameter version of the bilinear multiplier theorem of Coifman and Meyer. In fact, we prove a weighted version of the result, and recover, apart from some $L^{\infty}$ endpoints, the full range of mixed-norm estimates proved by Benea-Muscalu.

We use the theory of layer potentials to study boundary value problems on SKT domains, a family of domains introduced by S. Semmes, C. Kenig and T. Toro. We consider the case of elliptic systems with constant coefficients in which the boundary data belongs to $L^p(\partial \Omega,w)$, where $w \in A_p(\partial \Omega)$ is a Muckenhoupt weight. This extends previous results by S. Hofmann, M. Mitrea and M. Taylor. This approach relies on the invertibility of layer potentials defined on the boundary of the domain and therefore several tools are developed for this purpose, such as a quantitative interpolation theorem for compact operators. Moreover, these techniques can also be used for the cases of boundary data in other spaces, such as variable Lebesgue spaces or rearrangement-invariant Banach function spaces. This is joint work with J.M. Martell and M. Mitrea.

The study of minimal energy points (or Fekete points) is an old subject with connections ranging from error correcting codes to the mathematical models for superconductors. In this talk I will discuss the use of random point processes to obtain well distributed points in terms of discrepancy, separation and expected energy.

Given an open interval $I \subseteq \mathbb{R}$ and a measurable function $p:I\to [1,+\infty)$, the variable Lebesgue space $L^{p(\cdot)}(I)$ is the subspace of measurable functions $f:I\to \mathbb{R}$ such that the following norm is finite \[ \Vert f \Vert_{L^{p(\cdot)}(I)}=\inf\left\{\lambda > 0: \int_I \left|\frac{f(x)}{\lambda}\right|^{p(x)} \mathrm{d}x\leq 1\right\}. \] The topological dual of this Banach space is perfectly known when $\displaystyle \Vert p\Vert_{L^\infty(I)} < \infty$. However, if $\displaystyle \Vert p\Vert_{L^\infty(I)}=\infty$, its description has been an open problem for years.

In this talk, we are going to discuss some recent approaches that give a better understanding of the phenomena beyond it.

This a joint work with A. Amenta (TUDELFT), J.M. Conde-Alonso (Brown U.), and D. Cruz-Uribe (U. Alabama).

Let $\Phi_1 , \Phi_2 , \Phi_3$ be Young functions and $L^{\Phi_1}(\mathbb{R})$, $L^{\Phi_2}(\mathbb{R})$, $L^{\Phi_3}(\mathbb{R})$ be corresponding Orlicz spaces.

We say that a bounded function $m(\xi,\eta)$ defined on $\mathbb{R}\times \mathbb{R}$ is a bilinear multiplier on $\mathbb{R}$ of type $(\Phi_1,\Phi_2,\Phi_3)$ if \[ B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(\xi) \hat{g}(\eta)m(\xi,\eta)e^{2\pi i < \xi+\eta, x > }d\xi d\eta \] is a bounded bilinear operator from $L^{\Phi_1}(\mathbb{R}) \times L^{\Phi_2}(\mathbb{R})$ to $L^{\Phi_3}(\mathbb{R})$. We denoted by $BM_{(\Phi_1,\Phi_2,\Phi_3)}(\mathbb{R})$ the space of all bilinear multipliers of type $(\Phi_1,\Phi_2,\Phi_3)$.

In this talk we investigate some properties of the class $BM_{(\Phi_1,\Phi_2,\Phi_3)}(\mathbb{R})$ and will give some examples of bilinear multipliers. We will focus on the case $m(\xi,\eta)=M(\xi-\eta) \in BM_{(\Phi_1,\Phi_2,\Phi_3)}(\mathbb{R})$. Also, in the special case of this study we obtain some of the the known results for Lebesgue spaces.

This is a work in progress with Oscar Blasco.

*A.Osancliol is supported by "The Scientific and Technological Research Council of Turkey" TUBITAK-BIDEB grant no 1059B191600535.

It will be described the radial weights $\omega$ on the unit disc $\mathbb{D}$ such that the Littlewood-Paley formula \begin{equation*} \|f\|_{A^p_\omega}^p\asymp\int_{\mathbb{D}}|f^{(k)}(z)|^p(1-|z|)^{kp}\omega(z)\,dA(z)+\sum_{j=0}^{k-1}|f^{(j)}(0)|^p. \end{equation*} holds for any analytic function on $\mathbb{D}$, $0 < p < \infty$ and $k\in\mathbb{N}$. Moreover, if we are on time, related results on weighted Bergman projections will be presented.

These results are part of an ongoing work together with Jouni Rättyä.

In this talk we will see the continuity on modulation spaces $M^{p,q}$ of Fourier multipliers with symbols of the type $e^{i\mu(\xi)}$, for some real-valued function $\mu(\xi)$ called phase. A number of results are known, assuming that the derivatives of order $\geq 2$ of the phase are bounded or, more generally, that its second derivatives belong to the Sjöstrand class $M^{\infty,1}$. Here we extend those results, by assuming that the second derivatives lie in the bigger Wiener amalgam space $W( \mathcal{F}L^1,L^\infty)$. In particular, they could have stronger oscillations at infinity such as $\cos |\xi|^2$. In fact, the main result presented in this talk deals with the more general case of possibly unbounded second derivatives. In that case, we have boundedness on weighted modulation spaces with a sharp loss of derivatives.

Girela-Sarrión has recently studied the problem of estimating the maximal Cauchy transform in terms of the Cauchy transform itself in the context of smooth Jordan curves or Lipschitz graphs. If $T$ denotes the Cauchy transform, $T_*$ the maximal Cauchy transform and M the Hardy-Littlewood maximal operator with respect to the arc-length measure, one wants to understand under what conditions the Cotlar-type inequality $T_*(f)\leq M^2(Tf)$ holds, $M^2$ being the iteration of $M$. Girela-Sarrión showed that the previous inequality fails in the presence of an angle and gave a sufficient condition for its validity in terms of the smoothness of the curve. In this talk we discuss, under the background assumption of asymptotic quasi-conformality, a characterization of the curves such that the aforementioned inequality holds. We also provide an example of a curve which presents a critical behavior.

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic (in the Gromov sense) if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. We denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, \emph{i.e.}, $\delta(X):=\inf\{\delta\geq 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}\,. $ The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this paper some results are presented when studying the hyperbolicity constant $\delta(G)$ of a graph $G$ under certain operations, obtaining a new graph, such as the subdivision of edges, $S(G)$, the graph line, $\mathcal{L}(G)$, and the total graph, $T(G)$, among others. The results in this work are equalities and inequalities, such as, $\delta(S(G))/2= \delta(G) \leq \delta(\mathcal{L}(G)) \leq \delta (T(G))$ and $\delta(\mathcal{L}(G)) \leq \delta(Q(G)) \leq \delta(\mathcal{L}(G)) +1/2$.

This a joint work with J.M. Rodríguez (UC3M), and J.M. Sigarreta (UAGRO).

In this talk we deal with variable exponent local Hardy spaces related with a non-negative self-adjoint operator $L$ whose associated heat semigroup $\{e^{-tL}\}_{t > 0}$ satisfies Gaussian estimates. In particular we show a molecular characterization which allows us to get relations between our variable local Hardy spaces and their global counterpart.

Joint work with V. Almeida (ULL), J.J. Betancor (ULL), and E. Dalmasso (CONICET).

We review classical results concerning the bounds of the Hardy-Littlewood maximal operator on weighted Lorentz spaces and discuss the analogous bounds for the pointwise product of such operators, providing several applications for them. A new Hölder-type inequality for Lorentz spaces is used. This is joint work with Prof. María J. Carro.

In this paper we show that almost-greedy bases are equivalent to semi-greedy bases (notion introduced by Dilworth, Kalton and Kutzarova in 2003) in the context of Schauder bases in Banach spaces.

This property was only known under the additional restriction $|s| < 1/2$, in which case the basis is unconditional. The result complements earlier work by Seeger and Ullrich, which showed that $|s| < 1/2$ is necessary for unconditionality, and implies that the Haar system is a conditional basis in an open subset of the $(s,p)$ indices.

The results extend to the class of Triebel-Lizorkin $F(s,p,q)$ in $\mathbb{R}^d$, with $0< p,q < \infty$, establishing the Schauder basis property in the range $\max\{ d(1/p-1), 1/p-1 \} < s < \min \{1,1/p \}$, which is optimal except perhaps at the end-points.

We introduce the Hölder-Zygmund spaces associated to the parabolic Hermite operator $\mathcal{L}= \partial_t- \Delta_x+|x|^2$. The definitions are given through pointwise conditions. These spaces are natural in the regularity analysis of some differential operators.

We give an alternative integral definition of the mentioned spaces by using its Poisson semigroup, $e^{-y\sqrt{\mathcal{L}}}$. For $\alpha>0$, we define the spaces \[ \Lambda^\alpha_{\mathcal{L}} =\left\{f: \:f\in L^\infty(\mathbb{R}^{n+1})\:\; {\rm and} \:\; \left\|\partial_y^k e^{-y\sqrt{{\mathcal{L}}}} f \right\|_{L^\infty(\mathbb{R}^{n+1})}\leq C_k y^{-k+\alpha},\;\: {\rm with }\, k=[\alpha]+1, y>0 \right\}, \] with the obvious norm. It is shown that these spaces coincide with the ones defined via the pointwise condition.

Moreover, we show that the fractional powers $\mathcal{L}^{\pm \beta}$ are well defined in these spaces and satisfy the following inequalities: \begin{align*} \alpha, \beta >0, \quad \|\mathcal{L}^{-\beta} f\|_{\Lambda^{\alpha+2\beta}_{\mathcal{L}}} &\le C \|f\|_{\Lambda^\alpha_{\mathcal{L}}},\\ 0< 2\beta < \alpha, \quad \|\mathcal{L}^\beta f\|_{\Lambda_{\mathcal{L}}^{\alpha-2\beta}} &\le C \|f\|_{\Lambda^\alpha_{\mathcal{L}}}. \end{align*} In addition, it is also discussed the regularity of Riesz transforms, Bessel potentials and multipliers in these spaces. Parallel results are obtained for the Hermite operator $-\Delta_x+|x|^2$ in $\mathbb{R}^n$. The proofs use in a fundamental way the semigroup definition of the operators considered along the paper. The non-convolution structure of the operators produces an extra difficulty on the arguments.

This is a joint work with J.L. Torrea.

In this paper we prove new equalities involving the sequences $(a(n))_{n\ge 0}$ and $(b(n))_{n\ge 0}$ where \[ a(n):=\sum_{k=0}^{n+1}{n+k\choose n}^2, \qquad b(n):=\sum_{k=0}^{n+1}\frac{n+1-k}{ n+1}{n+k\choose n}^2\qquad n \ge 0, \] and the well-known Catalan numbers $(C_n)_{n\ge 0}$, given by \[ C_n={1\over n+1}{2n\choose n},\quad \ n\ge 0. \] We show that \begin{align*} 2(2n+1)a(n)-na(n-1)&= (4 + 21 n + 36 n^2 + 21 n^3) C_n^2, \qquad n\ge 1; \\ 2(2n+1)b(n)-nb(n-1)&= (7n^2+8n+2)C_{n}^2, \qquad n\ge 1. \end{align*} As a consequence, a new proof of the nice equality \[ ((n+1)C_n)^2= 3a(n-1)-2b(n), \qquad n\ge 1, \] which illustrates the intensive connection between these sequences $(a(n))_{n\ge 1 }$, $(b(n))_{n\ge 1}$ and Catalan numbers, is shown.

We find optimal decay estimates for the Poisson kernels associated with various Laguerre-type operators $L$. As a consequence we characterize the set of all the initial data $f$ whose Poisson integrals $\exp(-t L^{1/2})f$ converge a.e. to $f$. We also solve the 2-weight problem associated with the corresponding local maximal operators.