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Register a new userEndpoint sparse bounds for Walsh-Fourier multipliers of Marcinkiewicz type
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We prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain novel quantitative weighted norm inequalities for these operators. Among these, we establish the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<infty$ for Walsh-Littlewood-Paley square functions, and a restricted range for Marcinkiewicz multipliers. Zygmunds $L{(log L)^{{frac12}}}$ inequality is the core of our lacunary multi-frequency projection proof. We use the Walsh setting to avoid extra complications in the arguments.
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For $ 0< lambda < frac{1}2$, let $ B_{lambda }$ be the Bochner-Riesz multiplier of index $ lambda $ on the plane. Associated to this multiplier is the critical index $1 < p_lambda = frac{4} {3+2 lambda } < frac{4}3$. We prove a sparse bound for $ B_{lambda }$ with indices $ (p_lambda , q)$, where $ p_lambda < q < 4$. This is a further quantification of the endpoint weak $L^{p_lambda}$ boundedness of $ B_{lambda }$, due to Seeger. Indeed, the sparse bound immediately implies new endpoint weighted weak type estimates for weights in $ A_1 cap RH_{rho }$, where $ rho > frac4 {4 - 3 p_{lambda }}$.
We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for Hormanders theorem on the invertibility of homogeneous Fourier multipliers. Also, applications to the theory of Sobolev spaces will be given.
We prove the endpoint weak type estimate for square functions of Marcinkiewicz type with fractional integrals associated with non-isotropic dilations. This generalizes a result of C. Fefferman on functions of Marcinkiewicz type by considering fractional integrals of mixed homogeneity in place of the Riesz potentials of Euclidean structure.
Let $Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $mu_{Omega}$ be the higher-dimensional Marcinkiewicz integral defined by $$mu_Omega(f)(x)= Big(int_0^inftyBig|int_{|x-y|leq t}frac{Omega(x-y)}{|x-y|^{n-1}}f(y)dyBig|^2frac{dt}{t^3}Big)^{1/2}. $$ In this paper, the authors establish a bilinear sparse domination for $mu_{Omega}$ under the assumption $Omegain L^{infty}(S^{n-1})$. As applications, some quantitative weighted bounds for $mu_{Omega}$ are obtained.
For 1<p<infty and for weight w in A_p, we show that the r-variation of the Fourier sums of any function in L^p(w) is finite a.e. for r larger than a finite constant depending on w and p. The fact that the variation exponent depends on w is necessary. This strengthens previous work of Hunt-Young and is a weighted extension of a variational Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses weighted adaptation of phase plane analysis and a weighted extension of a variational inequality of Lepingle.
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