Jump inequalities are the $r=2$ endpoint of Lépingle's inequality for $r$-variation of martingales. Extending earlier work by Pisier and Xu we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators as well as to pass via sampling from $\mathbb{R}^{d}$ to $\mathbb{Z}^{d}$ for jump estimates for Fourier multipliers.
We prove strong jump inequalities for a large class of operators of Radon type in the discrete and ergodic theoretical settings. These inequalities are the $r=2$ endpoints of the $r$-variational estimates studied in arXiv:1512.07523.
In this paper we develop a general method for improving Jensen-type inequalities for convex and, even more generally, for piecewise convex functions. Our main result relies on the linear interpolation of a convex function. As a consequence, we obtain improvements of some recently established Young-type inequalities. Finally, our method is also applied to matrix case. In such a way we obtain improvements of some important matrix inequalities known from the literature.
In this paper, we establish jump and variational inequalities for the Calderón commutators, which are typical examples of non-convolution Calderón-Zygmund operators. For this purpose, we also show jump and variational inequalities for para-products and commutators from pseudo-differential calculus, which are of independent interest. New ingredients in the proofs involve identifying Carleson measures constructed from sequences of stopping times, in addition to many Littlewood-Paley type estimates with gradient.
In this paper, we systematically study weighted jump and variational inequalities for rough operators. More precisely, we show some weighted jump and variational inequalities for the families $\mathcal T:=\{T_\varepsilon\}_{\varepsilon>0}$ of truncated singular integrals and $\mathcal M_Ω:=\{M_{Ω,t}\}_{t>0}$ of averaging operators with rough kernels, which are defined respectively by $$ T_\varepsilon f(x)=\int_{|y|>\varepsilon}\frac{Ω(y')}{|y|^n}f(x-y)dy$$ and $$M_{Ω,t} f(x)=\frac1{t^n}\int_{|y|<t}Ω(y')f(x-y)dy, $$ where the kernel $Ω$ belongs to $L^q(\mathbf S^{n-1})$ for $q>1$.
Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density. The techniques combine methods from several complex variables (estimates for $\bar \partial$) and the theory of localized frames in general reproducing kernel Hilbert…
The aim of this paper is to present an abstract and general approach to jump inequalities in harmonic analysis. Our principal conclusion is the refinement of $r$-variational estimates, previously known for $r>2$, to end-point results for the jump quasi-seminorm corresponding to $r=2$. This is applied to the dimension-free results recently obtained by the first two authors in collaboration with Bourgain and Wróbel (arXiv:1708.04639 and arXiv:1804.07679), and also to operators of Radon type treated by Jones, Seeger, and Wright.
In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones {\it et al}. More precisely, we show some jump and variational inequalities for the families $\mathcal T:=\{T_\varepsilon\}_{\varepsilon>0}$ of truncated singular integrals and $\mathcal M:=\{M_t\}_{t>0}$ of averaging operators with rough kernels, which are defined respectively by $$ T_\varepsilon f(x)=\int_{|y|>\varepsilon}\frac{Ω(y')}{|y|^n}f(x-y)dy $$ and $$M_t f(x)=\frac1{t^n}\int_{|y|<t}Ω(y')f(x-y)dy, $$ where the kernel $Ω$ belongs to $L\log^+\!\!L(\ma…
Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems that related to integral estimates and regularity of solutions to the elliptic and/or parabolic equations. Since many of researchers who are working on this domain, the main motivation of our work here is to provide an important observation of $L^p$-boundedness property with respect to the interpolation inequalities. In this paper, from an idea of integral approximation theory and Sobolev, Morrey embeddings, we construct a nontrivial counte…
We prove an endpoint version of the uniform Sobolev inequalities in Kenig-Ruiz-Sogge [8]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [2]. We also prove restricted weak type Stein-Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.
Let $ϕ\in \mathscr{S}$ with $\intϕ(x)\, dx=1$, and define $$ϕ_t(x)=\frac{1}{t^n}ϕ(\frac{x}{t}),$$ and denote the function family $\{ϕ_t\ast f(x)\}_{t>0}$ by $Φ\ast f(x)$. Suppose that there exists a constant $C_1$ such that $$\sum_{t>0} |\hatϕ_t(x)|^2<C_1$$ for all $x\in \mathbb{R}^n$. Then (i) There exists a constant $C_2>0$ such that $$\|\mathscr{V}_2(Φ\ast f)\|_{L^p}\leq C_2\|f\|_{H^p},\;\;\frac{n}{n+1}<p\leq 1$$ for all $f\in H^p(\mathbb{R}^n)$, $\frac{n}{n+1}<p\leq 1$. (ii) The $λ$-jump operator $N_λ(Φ\ast f)$ satisfies $$\|λ[N_λ(Φ\ast f)]^{1/2}\|_{L^p}\leq C_3\|f\|_{H^p},\;\;\frac{n}{n+1…
In this paper we establish $L^p$-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.
In 2006 Carbery raised a question about an improvement on the naïve norm inequality $\|f+g\|_p^p \leq 2^{p-1}(\|f\|_p^p + \|g\|_p^p)$ for two functions in $L^p$ of any measure space. When $f=g$ this is an equality, but when the supports of $f$ and $g$ are disjoint the factor $2^{p-1}$ is not needed. Carbery's question concerns a proposed interpolation between the two situations for $p>2$. The interpolation parameter measuring the overlap is $\|fg\|_{p/2}$. We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all $…
Using C. Fefferman's embedding of a charge space in a measure space allows us to apply standard interpolation theorems to prove norm inequalities for Besicovitch almost periodic functions. This yields an analogue of Paley's Inequality for the Fourier coefficients of periodic functions.
In this work, we aim to study the action of composing by a rotation on the biparameter $\text{BMO}$ space in $\mathbb{R}^2$. This $\text{BMO}$ space is not preserved by a rotation since it relies on the structure of axis-parallel rectangles. We will quantify this fact by interpolation inequalities. One straightforward application of the interpolation inequalities is a boundedness property of directional Hilbert transforms.
We investigate the convergence of entire Lagrange interpolations and of Hermite interpolations of exponential type in weighted $L^p$-spaces on the real line. The weights are reciprocals of entire functions and depend on the type and may be viewed as smoothed versions of a target weight. The convergence statements are obtained from Marcinkiewicz inequalities with constants proportional to the type. For the special case of power weights we recover results of Rahman and Grozev and of Lubinsky.
In this note, we establish several interpolation inequalities in $\mathbb R^n$ in the Lebesgue spaces and Morrey spaces. By using the classical Calderon--Zygmund decomposition, we will reprove that $L^{p}(\mathbb R^n)\cap\mathrm{BMO}(\mathbb R^n)\subset L^{q}(\mathbb R^n)$ for all $q$ with $p<q<\infty$, where $1\leq p<\infty$. We also reprove that there exists a constant $C(p,q,n)$ depending on $p,q,n$ such that the following inequality \begin{equation*} \|f\|_{L^q}\leq C(p,q,n)\cdot\big(\|f\|_{L^p}\big)^{p/q}\cdot\big(\|f\|_{\mathrm{BMO}}\big)^{1-p/q} \end{equation*} holds for all $f\in L^{p}…
