It is proved that there does not exist any non zero function in Lp(Rn)
with 1≤p≤2n/α if its Fourier transform is supported by a set of
finite packing α-measure where 0<α<n. It is shown that the
assertion fails for p>2n/α. The result is applied to prove Lp
Wiener-Tauberian theorems for Rn and M(2)
Suppose μ is an α-dimensional fractal measure for some
0<α<n. Inspired by the results proved by R. Strichartz in 1990, we
discuss the Lp-asymptotics of the Fourier transform of fdμ by estimating
bounds of
L→∞liminfLk1∫∣ξ∣≤L∣fdμ(ξ)∣pdξ, for f∈Lp(dμ) and 2<p<2n/α. In a
different direction, we prove a Hardy type inequality, that is,
∫(μ(Ex))2−p∣f(x)∣pdμ(x)≤CL→∞liminfLn−α1∫BL(0)∣fdμ(ξ)∣pdξ where 1≤p≤2 and
Ex=E∩(−∞,x1]×(−∞,x2]...(−∞,xn] for
x=(x1,...xn)∈Rn generalizing the one dimensional results proved by
Hudson and Leckband in 1992
In this paper we study weighted estimates for the multi-frequency
ω−Calder\'{o}n-Zygmund operators T associated with the frequency set
Θ={ξ1,ξ2,…,ξN} and modulus of continuity ω
satisfying the usual Dini condition. We use the modern method of domination by
sparse operators and obtain bounds ∥T∥Lp(w)→Lp(w)≲N∣r1−21∣[w]Ap/rmax(1,p−r1),1≤r<p<∞, for the exponents of N and Ap/r characteristic
[w]Ap/r