In this paper we provide some quantitative mixed-type estimates assuming conditions that imply that $uv\in A_{\infty}$ for Calderón-Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced by Domingo-Salazar, Lacey and Rey and extended in works by Lerner, Ombrosi and the second author and Li, Perez, the second author and Roncal.
We give a simpler proof of a result of Holland concerning a mixed arithmetic-geometric mean inequality. We also prove a result of mixed mean inequality involving the symmetric means.
We devote this note to correct an estimate concerning mixed inequalities for the generalized maximal function $M_Φ$, when certain properties of the associated Young function $Φ$ are assumed. Although the obtained estimates turn out to be slightly different, they are good extensions of mixed inequalities for the classical Hardy-Littlewood maximal functions $M_r$, with $r\geq 1$. They also allow us to obtain mixed estimates for the generalized fractional maximal operator $M_{γ,Φ}$, when $0<γ<n$ and $Φ$ is an $L\log L$ type function.
We improve on several mixed weak type inequalities both for the Hardy-Littlewood maximal function and for Calderón-Zygmund operators. These type of inequalities were considered by Muckenhoupt and Wheeden and later on by Sawyer estimating the $L^{1, \infty}(uv)$ norm of $v^{-1}T(fv)$ for special cases. The emphasis is made in proving new and more precise quantitative estimates involving the $A_p$ or $A_\infty$ constants of the weights involved.
In this paper we provide mixed weak type inequalities generalizing previous results in an earlier work by Caldarelli and the second author and also in the spirit of earlier results by Lorente, Martell, Pérez and Riveros. One of the main novelties is that, besides obtaining estimates in the euclidean setting, results are provided as well in spaces of homogeneous type, being the first mixed weak type estimates that we are aware of in that setting.
In this article we consider a method of proving a class of inequalities of the form (1). The method is based on the precise approximations of the sine and cosine functions by Maclaurin polynomials of given order. By using this method we present new proofs of some inequalities from the articles C.-P. Chen, W.-S. Cheung [J. Inequal. Appl. 2012:72 (2012)] and Z.-J. Sun, L. Zhu [ISRN Math. Anal. (2011)].
We prove mixed $A_p$-$A_r$ inequalities for several basic singular integrals, Littlewood-Paley operators, and the vector-valued maximal function. Our key point is that $r$ can be taken arbitrary big. Hence such inequalities are close in spirit to those obtained recently in the works by T. Hytönen and C. Pérez, and M. Lacey. On one hand, the "$A_p$-$A_{\infty}$" constant in these works involves two independent suprema. On the other hand, the "$A_p$-$A_r$" constant in our estimates involves a joint supremum but of a bigger expression. We show in simple examples that both such constants are incom…
The aim of this paper is to obtain mixed weak-type inequalities for multilinear fractional operators, extending results by F. Berra, M. Carena and G. Pradolini \cite{BCP}. We prove that, under certain conditions on the weights, there exists a constant $C$ such that $$\Bigg\| \frac{\mathcal G_α(\vec f \,)}{v}\Bigg\|_{L^{q, \infty}(νv^q)} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(u_i)}},$$ where $\mathcal G_α(\vec f \,)$ is the multilinear maximal function $\mathcal M_α(\vec f\,)$ that was introduced by K. Moen in \cite{M} or the multilineal fractional integral $\mathcal I_α(\vec f \,)$. As an applica…
In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,...,w_m)$ and $ν= w_1^\frac{1}{m}...w_m^\frac{1}{m}$, the main result of the paper sentences that under different conditions on the weights we can obtain $$\Bigg\| \frac{T(\vec f\,)(x)}{v}\Bigg\|_{L^{\frac{1}{m}, \infty}(νv^\frac{1}{m})} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(w_i)}}, $$ where $T$ is a multilinear Calderón-Zygmund operator. To obtain this result we first prove it for the $m$-fold product of the Hardy-Littlewood maximal operator…
This paper gives the pointwise sparse dominations for variation operators of singular integrals and commutators with kernels satisfying the $L^r$-Hörmander conditions. As applications, we obtain the strong type quantitative weighted bounds for such variation operators as well as the weak-type quantitative weighted bounds for the variation operators of singular integrals and the quantitative weighted weak-type endpoint estimates for variation operators of commutators, which are completely new even in the unweighted case. In addition, we also obtain the local exponential decay estimates for such…
In this article we prove mixed inequalities for maximal operators associated to Young functions, which are an improvement of a conjecture established in \cite{Berra}. Concretely, given $r\geq 1$, $u\in A_1$, $v^r\in A_\infty$ and a Young function $Φ$ with certain properties, we have that inequality \[uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{M_Φv(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\] holds for every positive $t$. The involved operator $\frac{M_Φ(fv)(x)}{M_Φv(x)}$ seems to be an adequate extension when $v^r\in A_\infty$, since wh…
We give Feffermain-Stein type inequalities related to mixed estimates for Calderón-Zygmund operators. More precisely, given $δ>0$, $q>1$, $φ(z)=z(1+\log^+z)^δ$, a nonnegative and locally integrable function $u$ and $v\in \mathrm{RH}_\infty\cap A_q$, we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T(fv)(x)|}{v(x)}>t\right\}\right)\leq \frac{C}{t}\int_{\mathbb{R}^n}|f|\left(M_{φ, v^{1-q'}}u\right)M(Ψ(v))\] holds with $Ψ(z)=z^{p'+1-q'}\mathcal{X}_{[0,1]}(z)+z^{p'}\mathcal{X}_{[1,\infty)}(z)$, for every $t>0$ and every $p>\max\{q,1+1/δ\}$. This inequality provides a more ge…
We prove mixed inequalities for the generalized maximal operator $M_Φ$ when the function $v$ is a radial power function that fails to be locally integrable. Concretely, let $u$ be a weight, $v(x)=|x|^β$ with $β<-n$ and $r\geq 1$. If $Φ$ is a Young function with certain properties, then the inequality \[uv^r\left(\left\{x\in\mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)v^r(x)Mu(x)\,dx\] holds for every $t>0$ and every bounded function. This improves a similar mixed estimate proved in \cite{BCP-M}. As an application, we give mixed es…
For every symmetric mean $\mathscr{M} \colon \bigcup_{n=1}^\infty I^n \to I$ (where $I$ an interval) and a nonzero function $W \colon \{1,\dots,n\} \to \mathbb{N} \cup \{0\}$, define an $n$-variable mean by $$\mathscr{M}_W(x):=\mathscr{M}\big(\underbrace{x_1,\dots,x_1}_{W(1)\text{-times}},\dots,\underbrace{x_n,\dots,x_n}_{W(n)\text{-times}}\big) \text{ for }x=(x_1,\dots,x_n) \in I^n.$$ Given two symmetric means $\mathscr{M},\,\mathscr{N} \colon \bigcup_{n=1}^\infty I^n \to I$ satisfying the so-called Ingham--Jessen inequality and some nonzero functions $F_1,\dots,F_k$, $G_1,\dots,G_l \colon \{…
We study mixed weighted weak-type inequalities for families of functions, which can be applied to study classical operators in harmonic analysis. Our main theorem extends the key result from D. Cruz-Uribe, J.M. Martell and C. Perez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not., V. 30, 2005, 1849-1871.
We prove mixed inequalities for commutators of Calderón-Zygmund operators (CZO) with multilinear symbols. Concretely, let $m\in\mathbb{N}$ and $\mathbf{b}=(b_1,b_2,\dots, b_m)$ be a vectorial symbol such that each component $b_i\in \mathrm{Osc}_{\mathrm{exp}\, L^{r_i}}$, with $r_i\geq 1$. If $u\in A_1$ and $v\in A_\infty(u)$ we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T_\mathbf{b}(fv)(x)|}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\|\mathbf{b}\|\frac{|f(x)|}{t}\right)u(x)v(x)\,dx\] holds for every $t>0$, where $Φ(t)=t(1+\log^+t)^r$, with $1/r=\sum_{i=1}^…
Let $f\in \ell^2(\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\frac{1}{N}\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \ell ^{p}$-improving estimate, for $ 3/2 < p \leq 2$: \begin{equation*} N ^{-2/p'} \lVert A_N f \rVert _{ p'} \lesssim N ^{-2/p} \lVert f\rVert _{\ell ^{p}}, \end{equation*} provided $ f$ is supported in some interval of length $ N ^2 $, and $ p' =\frac{p} {p-1}$ is the conjugate index. The inequality above fails for $ 1< p < 3/2$. The maximal function $ A f = \sup _{N\geq 1} |A_Nf| $ satisfies a similar sparse b…
Let $ T_{P } f (x) = \int e ^{i P (y)} K (y) f (x-y) \; dy $, where $ K (y)$ is a smooth Calderón-Zygmund kernel on $ \mathbb R ^{n}$, and $ P$ be a polynomial. We show that there is a sparse bound for the bilinear form $ \langle T_P f, g \rangle$. This in turn easily implies $ A_p $ inequalities. The method of proof is applied in a random discrete setting, yielding the first weighted inequalities for operators defined on sparse sets of integers.
We prove a sparse bound for the $m$-sublinear form associated to vector-valued maximal functions of Fefferman-Stein type. As a consequence, we show that the sparse bounds of multisublinear operators are preserved via $\ell^r$-valued extension. This observation is in turn used to deduce vector-valued, multilinear weighted norm inequalities for multisublinear operators obeying sparse bounds, which are out of reach for the extrapolation theory recently developed by Cruz-Uribe and Martell. As an example, vector-valued multilinear weighted inequalities for bilinear Hilbert transforms are deduced fr…
The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities by reducing to polynomial inequalities. Finally, we show the great applicability of this algorithm and as examples, we use it to analyze some new rational (Pade) approximations of the function $\cos^2(x)$, and to improve a class of inequalities by Z.-H. Yang. The results of our analysis could be implemented by means of an automated proof assistant, so our work is a contribution to the library of automatic support tools for proving various analytic inequalities.
In this paper we refine the recent sparse domination of the integrated $p = 2$ matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A. Volberg to prove a pointwise sparse domination of general matrix weighted dyadic square functions. We then use this to prove sharp two matrix weighted strong type inequalities for matrix weighted dyadic square functions when $1 < p \leq 2$.
We prove Bloom type two-weight inequalities for commutators of multilinear singular integral operators including Calderón-Zygmund operators and their dyadic counterparts. Such estimates are further extended to a general higher order multilinear setting. The proof involves a pointwise sparse domination of multilinear commutators.
For polynomial $ P (x,y)$, and any Calderón-Zygmund kernel, $K$, the operator below satisfies a $ (1,r)$ sparse bound, for $ 1< r \leq 2$. $$ \sup _{ε>0} \Bigl\lvert \int_{|y| > ε} f (x-y) e ^{2 πi P (x,y) } K(y) \; dy \Bigr\rvert $$ The implied bound depends upon $ P (x,y)$ only through the degree of $ P$. We derive from this a range of weighted inequalities, including weak type inequalities on $ L ^{1} (w)$, which are new, even in the unweighted case. The unweighted weak-type estimate, without maximal truncations, is due to Chanillo and Christ (1987).
Let $\vec{p}\in(0,1]^n$ be a $n$-dimensional vector and $A$ a dilation. Let $H_A^{\vec{p}}(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H_{A}^{\vec{p}}(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f\in H_A^{\vec{p}}(\mathbb{R}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. Moreover, the function $F$ can be controlled pointwisely by the product of the Hardy spac…
In this paper, we establish several new anisotropic Hardy-Sobolev inequalities in mixed Lebesgue spaces and mixed Lorentz spaces, which covers many known corresponding results. As an application, this type of inequalities allows us to generalize some regularity criteria of the 3D axisymmetric Navier-Stokes equations.
