Fractional Maximal Functions Research Articles

Sobolev type inequalities for fractional maximal functions and Green potentials in half spaces

In this paper, we study Sobolev type inequalities for fractional maximal functions in the half space. We also discuss Sobolev type inequalities for Green potentials in the half space.

Global weighted Lorentz estimates for parabolic equations with measure via strong fractional maximal functions

Global weighted Lorentz estimates for parabolic equations with measure via strong fractional maximal functions

Boundedness of weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Cesàro function spaces

The boundedness of the weighted iterated Hardy-type operators \(T_{u,b}\) and \(T_{u,b}^*\) involving suprema from weighted Lebesgue space \(L_p(v)\) into weighted Cesàro function spaces \({\operatorname{Ces}}_{q}(w,a)\) are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator \(R_u\) from \(L^p(v)\) into \({\operatorname{Ces}}_{q}(w,a)\) on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator \(P_{u,b }\) from \(L^p(v)\) into \({\operatorname{Ces}}_{q}(w,a)\) on the cone of monotone non-increasing functions. Under additional condition on \(u\) and \(b\), we are able to characterize the boundedness of weighted iterated Hardy-type operator \(T_{u,b}\) involving suprema from \(L^p(v)\) into \({\operatorname{Ces}}_q(w,a)\) on the cone of monotone non-increasing functions. At the end of the paper, as an application of obtained results, we calculate the norm of the fractional maximal function \(M_{\gamma}\) from \(\Lambda^p(v)\) into \(\Gamma^q(w)\).

Sobolev regularity for commutators of the fractional maximal functions

In this paper the Sobolev regularity properties are investigated of the commutators of fractional maximal functions, both in the global and local case. Some new bounds for the derivatives of the above commutators will be established. As several applications, the boundedness for these operators in Sobolev spaces as well as the bounds of these operators on the Sobolev spaces with zero boundary values in the local setting are obtained.

Level-set inequalities on fractional maximal distribution functions and applications to regularity theory

The aim of this paper is to establish an abstract theory based on the so-called fractional-maximal distribution functions (FMDs). From the basic ideas introduced in [1], we develop and prove some abstract results related to the level-set inequalities and norm-comparisons by using the language of such FMDs. Particularly interesting is the applicability of our approach that has been shown in regularity and Calderón-Zygmund type estimates. In this paper, due to our research experience, we will establish global regularity estimates for two types of general quasilinear problems (problems with divergence form and double obstacles), via fractional-maximal operators and FMDs. The range of applications of these abstract results is large. Apart from these two examples of the regularity theory for elliptic equations discussed, it is also promising to indicate further possible applications of our approach for other special topics.

Riesz Potential and Maximal Function for Dunkl transform

We study weighted (Lp, Lq)-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy–Littlewood–Sobolev type inequality and weighted week (L1, Lq) estimate. We find a sharp constant in the weighted Lp-inequality, generalizing the results of W. Beckner and S. Samko.

Regularity of the centered fractional maximal function on radial functions

We study the regularity properties of the centered fractional maximal function Mβ. More precisely, we prove that the map f↦|∇Mβf| is bounded and continuous from W1,1(Rd) to Lq(Rd) in the endpoint case q=d/(d−β) if f is a radial function. For d=1, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.

Weak differentiability for fractional maximal functions of general Lp functions on domains

Let Ω⊂Rn be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to Ω maps Lp(Ω)→W1,p(Ω) for all p>1, when the smoothness index α≥1. In particular, the results are valid in the range p∈(1,n/(n−1)] that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting.

Two‐weight norm inequalities for fractional maximal functions and fractional integral operators on weighted Morrey spaces

AbstractIn this paper, we consider weighted norm inequalities for fractional maximal operators and fractional integral operators. For suitable weights, we prove the two‐weight norm inequalities for both operators on weighted Morrey spaces.

Endpoint Sobolev Continuity of the Fractional Maximal Function in Higher Dimensions

Abstract We establish continuity mapping properties of the noncentered fractional maximal operator $M_{\beta }$ in the endpoint input space $W^{1,1}({\mathbb R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. More precisely, we prove that for $q=d/(d-\beta )$ the map $f \mapsto |\nabla M_\beta f|$ is continuous from $W^{1,1}({\mathbb R}^d)$ to $L^{q}({\mathbb R}^d)$ for $ 0 < \beta < 1$ if $f$ is radial and for $1 \leq \beta < d$ for general $f$. The results for $1\leq \beta < d$ extend to the centered counterpart $M_\beta ^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_\beta ^c$ implies its continuity.

Double phase image restoration

In this paper we explore the potential of the double phase functional in an image processing context. To this end, we study minimizers of the double phase energy for functions with bounded variation and show that this energy can be obtained by Γ-convergence or relaxation of regularized functionals. A central tool is a capped fractional maximal function of the derivative of BV functions.

New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data

This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions:{div(A(x,∇u))=div(|F|p−2F)inΩ,u=σon∂Ω, where Ω⊂Rn (n≥2), the nonlinearity A is a monotone Carathéodory vector valued function defined on W01,p(Ω) for p>1 and the p-capacity uniform thickness condition is imposed on the complement of our bounded domain Ω. Moreover, for given data F∈Lp(Ω;Rn), the problem is set up with general Dirichlet boundary data σ∈W1,p(Ω). In this paper, the optimal good-λ type bounds technique is applied to prove some results of fractional maximal estimates for gradient of solutions. And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works [45,52,53] and references therein.

Weighted boundedness of multilinear maximal function using Dirac deltas

In this article we extend a method of Miguel de Guzman involving boundedness properties of maximal functions using Dirac deltas to multilinear setting. This method involves estimating maximal functions over finite linear combination of Dirac deltas. As an application, we obtain end-point weighted boundedness of the multilinear Hardy–Littlewood fractional maximal function with respect to multilinear weights.

Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

Let 0<alpha<n and M_{alpha} be the fractional maximal function. The nonlinear commutator of M_{alpha} and a locally integrable function b is given by [b,M_{alpha}](f)=bM_{alpha}(f)-M_{alpha}(bf). In this paper, we mainly give some necessary and sufficient conditions for the boundedness of [b,M_{alpha}] on variable Lebesgue spaces when b belongs to Lipschitz or mathit{BMO}({mathbb{R}}^{n}) spaces, by which some new characterizations for certain subclasses of Lipschitz and mathit{BMO}({mathbb{R}}^{n}) spaces are obtained.

Matrix weighted Poincare inequalities and applications to degenerate elliptic systems

We prove Poincar\'e and Sobolev inequalities in matrix A${}_p$ weighted spaces. We then use these Poincar\'e inequalities to prove existence and regularity results for degenerate systems of elliptic equations whose degeneracy is governed by a matrix A${}_p$ weight. Such results parallel earlier results by Fabes, Kenig, and Serapioni for a single degenerate equation governed by a scalar A${}_p$ weight. In addition, we prove Cacciopoli and reverse H\"older inequalities for weak solutions of the degenerate systems. As a means to prove the Poincar\'e inequalities we prove that the Riesz potential and fractional maximal function operators are bounded on matrix weighted $L^p$ spaces and go on to develop an entire matrix A${}_{p, q}$ theory.

Regularity of one-sided multilinear fractional maximal functions

AbstractIn this paper we introduce and investigate the regularity properties of one-sided multilinear fractional maximal operators, both in continuous case and in discrete case. In the continuous setting, we prove that the one-sided multilinear fractional maximal operators$\mathfrak{M}_\beta^{+}\; \text{and}\, \mathfrak{M}_\beta^{-}$map W1,p1 (ℝ)×· · ·×W1,pm (ℝ) into W1,q(ℝ) with 1 &lt; p1, … , pm &lt; ∞, 1 ≤ q &lt; ∞ and $1/q= \sum_{i=1}^m1/p_i-\beta$, boundedly and continuously. In the discrete setting, we show that the discrete one-sided multilinear fractional maximal operators are bounded and continuous from ℓ1(ℤ)×· · ·×ℓ1(ℤ) to BV(ℤ). Here BV(ℤ) denotes the set of functions of bounded variation defined on ℤ. Our main results represent significant and natural extensions of what was known previously.

Regularity of fractional maximal functions through Fourier multipliers

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions n≥2. We also show that the spherical fractional maximal function maps Lp into a first order Sobolev space in dimensions n≥5.

Commutators of Some Maximal Functions with Lipschitz Function on Orlicz Spaces

Let $$0\le \alpha <n$$ , $$[b,M_{\alpha }]$$ and $$[b,M^{\sharp }]$$ be the nonlinear commutators of the fractional maximal function $$M_{\alpha }$$ and the sharp maximal function $$M^{\sharp }$$ with a locally integrable function b. In this note, we give necessary and sufficient conditions for the boundedness of $$[b,M_{\alpha }]$$ and $$[b,M^{\sharp }]$$ on Orlicz spaces when the symbol b belongs to Lipschitz spaces, by which some new characterizations of non-negative Lipschitz functions (in terms of the Orlicz norm) are obtained.

On the regularity of one-sided fractional maximal functions

Abstract In this paper we investigate the regularity properties of one-sided fractional maximal functions, both in continuous case and in discrete case. We prove that the one-sided fractional maximal operators ℳ β + $ \mathcal{M}_{\beta}^{+} $ and ℳ β - $ \mathcal{M}_{\beta}^{-} $ map W 1 , p ⁢ ( ℝ ) $ W^{1,p}(\mathbb{R}) $ into W 1 , q ⁢ ( ℝ ) $ W^{1,q}(\mathbb{R}) $ with 1 &lt;p &lt;∞, 0≤β&lt;1/p and q=p/(1-pβ), boundedly and continuously. In addition, we also obtain the sharp bounds and continuity for the discrete one-sided fractional maximal operators M β + $ M_{\beta}^{+} $ and M β - $ M_{\beta}^{-} $ from ℓ 1 ⁢ ( ℤ ) $ \ell^{1}(\mathbb{Z}) $ to BV ⁢ ( ℤ ) $ {\rm BV}(\mathbb{Z}) $ . Here BV ⁢ ( ℤ ) $ {\rm BV}(\mathbb{Z}) $ denotes the set of all functions of bounded variation defined on ℤ. The results we obtained represent significant and natural extensions of what was known previously.

Two-Weight Weak Type Inequality for Modified Fractional Maximal Function Defined with Respect to Non-Doubling Measure and Some Applications in Non-standard Lebesgue Spaces

Let $$ \mu $$ a nonnegative Radon measure on $$ {\mathbb{R}}^{d} $$ ; $$ p,q,\gamma ,k $$ real numbers; $$ M_{\mu ,k}^{\gamma } $$ a fractional maximal operator; $$ A_{p,q}^{\gamma ,k} \left( \mu \right) $$ a Muckenhoupt condition associated to $$ \mu $$ ; $$ L^{p( \cdot )} ({\mathbb{R}}^{d} , \mu ) $$ and $$ F(q, p,\alpha ,\mu )({\mathbb{R}}^{d} ) $$ two generalized Lebesgue spaces. The purpose of the present work is double: