scholarly journals Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

2016 ◽  
Vol 66 (4) ◽  
pp. 1079-1101
Author(s):  
Kristóf Szarvas ◽  
Ferenc Weisz
Keyword(s):  
Maximal Operator ◽  
Type Inequality ◽  
Strong Type ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Strong Type Inequality

scholarly journals Estimates of Fractional Integral Operators on Variable Exponent Lebesgue Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Canqin Tang ◽  
Qing Wu ◽  
Jingshi Xu
Keyword(s):  
Integral Operator ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Weak Type ◽  
Type Inequality ◽  
Integral Operators ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Variable Exponent Lebesgue Spaces

By some estimates for the variable fractional maximal operator, the authors prove that the fractional integral operator is bounded and satisfies the weak-type inequality on variable exponent Lebesgue spaces.

scholarly journals Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Joaquín Motos ◽  
María Jesús Planells ◽  
César F. Talavera
Keyword(s):  
Analytic Functions ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Previous Article ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Complemented Subspace ◽  
Open Set ◽  
Hörmander Spaces ◽  
Littlewood Maximal Operator

We show that the dual Bp·locΩ′ of the variable exponent Hörmander space Bp(·)loc(Ω) is isomorphic to the Hörmander space B∞c(Ω) (when the exponent p(·) satisfies the conditions 0<p-≤p+≤1, the Hardy-Littlewood maximal operator M is bounded on Lp(·)/p0 for some 0<p0<p- and Ω is an open set in Rn) and that the Fréchet envelope of Bp(·)loc(Ω) is the space B1loc(Ω). Our proofs rely heavily on the properties of the Banach envelopes of the p0-Banach local spaces of Bp(·)loc(Ω) and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for p(·)≡p, 0<p<1, are also given (e.g., all quasi-Banach subspace of Bploc(Ω) is isomorphic to a subspace of lp, or l∞ is not isomorphic to a complemented subspace of the Shapiro space hp-). Finally, some questions are proposed.

Weighted Inequalities for Hardy–Steklov Operators

2007 ◽  
Vol 59 (2) ◽  
pp. 276-295 ◽  
Author(s):  
A. L. Bernardis ◽  
F. J. Martín-Reyes ◽  
P. Ortega Salvador
Keyword(s):  
Weak Type ◽  
Type Inequality ◽  
Continuous Functions ◽  
Weighted Inequalities ◽  
Strong Type ◽  
Strong Type Inequality ◽  
Differentiability Properties

AbstractWe characterize the pairs of weights (v, w) for which the operator with s and h increasing and continuous functions is of strong type (p, q) or weak type (p, q) with respect to the pair (v, w) in the case 0 < q < p and 1 < p < ∞. The result for the weak type is new while the characterizations for the strong type improve the ones given by H. P. Heinig and G. Sinnamon. In particular, we do not assume differentiability properties on s and h and we obtain that the strong type inequality (p, q), q < p, is characterized by the fact that the functionbelongs to Lr(gqw), where 1/r = 1/q – 1/p and the supremum is taken over all c and d such that c ≤ x ≤ d and s(d) ≤ h(c).

scholarly journals Modular inequalities for the maximal operator in variable Lebesgue spaces

2018 ◽  
Vol 177 ◽  
pp. 299-311 ◽  
Author(s):  
David Cruz-Uribe ◽  
Giovanni Di Fratta ◽  
Alberto Fiorenza
Keyword(s):  
Maximal Operator ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Modular Inequalities
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