Singular Integrals and Function Spaces

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We show that Calderon-Zygmund operatorTf(x)=∫ K(x,y)f(y)dyis bounded on some Lipschitz and Sobolev spaces if T1 belongs to Lipschitz space. As a corollary of our theorem, we show that Calderon's commutatorC_Af(x)=∫(A(x)-A(y))/(x-y)^2f(y)dyis bounded on the spaces above.We show that discrete singular integral operatorTf(n)=Σ K(j)f(n-j)is bounded on discrete Besov space B^{0,1}_1(Z) if the kernel K(n) satisfies some regularity conditions.We define some maximal functions and weight classes on non-homogeneous spaces on R^n. And we show that these maximal functions are bounded on weighted L^p spaces.We define Carleson measure with respect to non-doubling measure, and we show that Poisson integral operator is bounded from L^p(R^n) to L^p(R^{n+1}+), (upper half space).