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In the Euclidean setting the Sobolev spaces $W^{alpha,p}cap L^infty$ are algebras for the pointwise product when $alpha>0$ and $pin(1,infty)$. This property has recently been extended to a variety of geometric settings. We produce a class of fractal examples where it fails for a wide range of the indices $alpha,p$.
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This paper is withdrawn. We found a mistake in Lemma 4.1
We consider the recovery of square-integrable signals from discrete, equidistant samples of their Gabor transform magnitude and show that, in general, signals can not be recovered from such samples. In particular, we show that for any lattice, one can construct functions in $L^2(mathbb{R})$ which do not agree up to global phase but whose Gabor transform magnitudes sampled on the lattice agree. These functions can be constructed to be either real-valued or complex-valued and have good concentration in both time and frequency.
We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.
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In this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of the classical quasidisks. After that, we also find some applications of their Sobolev extension property.
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