scholarly journals Strong-Type Inequality for Convolution with Square Root of the Poisson Kernel

2004 ◽  
Vol 75 (3/4) ◽  
pp. 542-552 ◽  
Author(s):  
I. N. Katkovskaya ◽  
V. G. Krotov
Keyword(s):  
Poisson Kernel ◽  
Type Inequality ◽  
Square Root ◽  
Strong Type ◽  
Strong Type Inequality

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 Approach regions for the square root of the Poisson kernel and bounded functions

1997 ◽  
Vol 55 (3) ◽  
pp. 521-527 ◽  
Author(s):  
P. Sjögren
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Monotone Function ◽  
Square Root ◽  
Almost Everywhere Convergence ◽  
Poisson Integrals ◽  
Almost Everywhere ◽  
Approach Region ◽  
Boundary Functions ◽  
Bounded Functions

If the Poisson kernel of the unit disc is replaced by its square root, it is known that normalised Poisson integrals of Lp boundary functions converge almost everywhere at the boundary, along approach regions wider than the ordinary non-tangential cones. The sharp approach region, defined by means of a monotone function, increases with p. We make this picture complete by determining along which approach regions one has almost everywhere convergence for L∞ boundary functions.

scholarly journals A Hardy space related to the square root of the Poisson kernel

2010 ◽  
Vol 199 (3) ◽  
pp. 207-225
Author(s):  
Jonatan Vasilis
Keyword(s):  
Hardy Space ◽  
Poisson Kernel ◽  
Square Root

scholarly journals Approach regions for $l^{p}$ potentials with respect to the square root of the Poisson kernel

2005 ◽  
Vol 96 (2) ◽  
pp. 243
Author(s):  
Martin Brundin
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Higher Dimensions ◽  
Square Root ◽  
Poisson Integrals ◽  
Boundary Functions

{If} one replaces the Poisson kernel of the unit disc by its square root, then normalised Poisson integrals of $L^{p}$ boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning ($1\leq p<\infty$) and Sjögren ($p=1$ and $p=\infty$). In this paper we present new and simplified proofs of these results. We also generalise the $L^{\infty}$ result to higher dimensions.

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