• Title/Summary/Keyword: weighted norm inequalities

Search Result 13, Processing Time 0.02 seconds

WEIGHTED ESTIMATES FOR CERTAIN ROUGH OPERATORS WITH APPLICATIONS TO VECTOR VALUED INEQUALITIES

  • Liu, Feng;Xue, Qingying
    • Journal of the Korean Mathematical Society
    • /
    • v.58 no.4
    • /
    • pp.1035-1058
    • /
    • 2021
  • Under certain rather weak size conditions assumed on the kernels, some weighted norm inequalities for singular integral operators, related maximal operators, maximal truncated singular integral operators and Marcinkiewicz integral operators in nonisotropic setting will be shown. These weighted norm inequalities will enable us to obtain some vector valued inequalities for the above operators.

INFINITE FINITE RANGE INEQUALITIES

  • Joung, Haewon
    • Korean Journal of Mathematics
    • /
    • v.18 no.1
    • /
    • pp.63-77
    • /
    • 2010
  • Infinite finite range inequalities relate the norm of a weighted polynomial over ${\mathbb{R}}$ to its norm over a finite interval. In this paper we extend such inequalities to generalized polynomials with the weight $W(x)={\prod}^{m}_{k=1}{\mid}x-x_k{\mid}^{{\gamma}_k}{\cdot}{\exp}(-{\mid}x{\mid}^{\alpha})$.

WEIGHTED Lp-BOUNDEDNESS OF SINGULAR INTEGRALS WITH ROUGH KERNEL ASSOCIATED TO SURFACES

  • Liu, Ronghui;Wu, Huoxiong
    • Journal of the Korean Mathematical Society
    • /
    • v.58 no.1
    • /
    • pp.69-90
    • /
    • 2021
  • In this paper, we prove weighted norm inequalities for rough singular integrals along surfaces with radial kernels h and sphere kernels Ω by assuming h ∈ △γ(ℝ+) and Ω ∈ ����β(Sn-1) for some γ > 1 and β > 1. Here Ω ∈ ����β(Sn-1) denotes the variant of Grafakos-Stefanov type size conditions on the unit sphere. Our results essentially improve and extend the previous weighted results for the rough singular integrals and the corresponding maximal truncated operators.

WEIGHTED ESTIMATES FOR CERTAIN ROUGH SINGULAR INTEGRALS

  • Zhang, Chunjie
    • Journal of the Korean Mathematical Society
    • /
    • v.45 no.6
    • /
    • pp.1561-1576
    • /
    • 2008
  • In this paper we shall prove some weighted norm inequalities of the form $${\int}_{R^n}\;|Tf(x)|^pu(x)dx\;{\leq}\;C_p\;{\int}_{R^n}\;|f(x)|^pNu(x)dx$$ for certain rough singular integral T and maximal singular integral $T^*$. Here u is a nonnegative measurable function on $R^n$ and N denotes some maximal operator. As a consequence, some vector valued inequalities for both T and $T^*$ are obtained. We shall also get a boundedness result of T on the Triebel-Lizorkin spaces.

WEIGHTED LEBESGUE NORM INEQUALITIES FOR CERTAIN CLASSES OF OPERATORS

  • Song, Hi Ja
    • Korean Journal of Mathematics
    • /
    • v.14 no.2
    • /
    • pp.137-160
    • /
    • 2006
  • We describe the weight functions for which Hardy's inequality of nonincreasing functions is satisfied. Further we characterize the pairs of weight functions $(w,v)$ for which the Laplace transform $\mathcal{L}f(x)={\int}^{\infty}_0e^{-xy}f(y)dy$, with monotone function $f$, is bounded from the weighted Lebesgue space $L^p(w)$ to $L^q(v)$.

  • PDF

WEIGHTED POISSON INTEGRAL IN THE UNIT DISC

  • Koo, Hyung-Woon;Park, Eun-Ui
    • Journal of applied mathematics & informatics
    • /
    • v.7 no.3
    • /
    • pp.1005-1015
    • /
    • 2000
  • In the unit disc, we find a sufficient condition to bound the Bergman norm by the weighted Poisson integral where the given weighted is $\mid$t$\mid$dt.

HOLOMORPHIC FUNCTIONS ON THE MIXED NORM SPACES ON THE POLYDISC

  • Stevic, Stevo
    • Journal of the Korean Mathematical Society
    • /
    • v.45 no.1
    • /
    • pp.63-78
    • /
    • 2008
  • We generalize several integral inequalities for analytic functions on the open unit polydisc $U^n={\{}z{\in}C^n||zj|<1,\;j=1,...,n{\}}$. It is shown that if a holomorphic function on $U^n$ belongs to the mixed norm space $A_{\vec{\omega}}^{p,q}(U^n)$, where ${\omega}_j(\cdot)$,j=1,...,n, are admissible weights, then all weighted derivations of order $|k|$ (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q ${\in}\;[1,\;{\infty})$ and when the order is equal to one. The equivalence of these conditions is given for all p, q ${\in}\;(0,\;{\infty})$ if ${\omega}_j(z_j)=(1-|z_j|^2)^{{\alpha}j},\;{\alpha}_j>-1$, j=1,...,n (the classical weights.) The main results here improve our results in Z. Anal. Anwendungen 23 (3) (2004), no. 3, 577-587 and Z. Anal. Anwendungen 23 (2004), no. 4, 775-782.