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Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by with D = x ∈ ℝ ²:|x| ≤ 1 and let be the convolution operator with the measure μ. Let be the decomposition of φ into irreducible factors. We show that if for each of degree 1, then the type set can be explicitly described as a closed polygonal region.
Let be the graph of the function defined by with 1< and let the measure on induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with is - bounded.
Let , 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let , and . Let φ₁,...,φₙ be real functions in such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on given by
,
where and dx denotes the Lebesgue measure on ℝⁿ. Let and let be the operator norm of from into , where the spaces are taken with respect to the Lebesgue measure. The type set is defined by
.
In the case for 1 ≤ i,k ≤ n we characterize the type set under...
Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from to for certain p,q. For m ≥ 6 the results are sharp except for some border points.
Let be real homogeneous functions in of degree , let and let be the Borel measure on given by
where denotes the Lebesgue measure on and . Let be the convolution operator and let
Assume that, for , the following two conditions hold: vanishes only at and . In this paper we show that if then is the empty set and if then is the closed segment with endpoints and . Also, we give some examples.
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