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The type set for homogeneous singular measures on ℝ ³ of polynomial type

E. FerreyraT. Godoy — 2006

Colloquium Mathematicae

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 μ ( E ) = D χ E ( x , φ ( x ) ) d x with D = x ∈ ℝ ²:|x| ≤ 1 and let T μ be the convolution operator with the measure μ. Let φ = φ e φ e be the decomposition of φ into irreducible factors. We show that if e i m / 2 for each φ i of degree 1, then the type set E μ : = ( 1 / p , 1 / q ) [ 0 , 1 ] × [ 0 , 1 ] : | | T μ | | p , q < can be explicitly described as a closed polygonal region.

Endpoint bounds for convolution operators with singular measures

E. FerreyraT. GodoyM. Urciuolo — 1998

Colloquium Mathematicae

Let S n + 1 be the graph of the function ϕ : [ - 1 , 1 ] n defined by ϕ ( x 1 , , x n ) = j = 1 n | x j | β j , with 1< β 1 β n , and let μ the measure on n + 1 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 μ is L p - L q bounded.

Convolution operators with anisotropically homogeneous measures on 2 n with n-dimensional support

E. FerreyraT. GodoyM. Urciuolo — 2002

Colloquium Mathematicae

Let α i , β i > 0 , 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let t x = ( t α x , . . . , t α x ) , t x = ( t β x , . . . , t β x ) and | | x | | = i = 1 n | x i | 1 / α i . Let φ₁,...,φₙ be real functions in C ( - 0 ) such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on 2 n given by μ ( E ) = χ E ( x , φ ( x ) ) | | x | | γ - α d x , where α = i = 1 n α i and dx denotes the Lebesgue measure on ℝⁿ. Let T μ f = μ f and let | | T μ | | p , q be the operator norm of T μ from L p ( 2 n ) into L q ( 2 n ) , where the L p spaces are taken with respect to the Lebesgue measure. The type set E μ is defined by E μ = ( 1 / p , 1 / q ) : | | T μ | | p , q < , 1 p , q . In the case α i β k for 1 ≤ i,k ≤ n we characterize the type set under...

Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³

E. FerreyraT. GodoyM. Urciuolo — 2004

Studia Mathematica

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 σ ( A ) = B χ A ( x , φ ( x ) ) d x 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 L p ( ³ ) to L q ( Σ , d σ ) for certain p,q. For m ≥ 6 the results are sharp except for some border points.

The type set for some measures on 2 n with n -dimensional support

E. FerreyraT. GodoyMarta Urciuolo — 2002

Czechoslovak Mathematical Journal

Let ϕ 1 , , ϕ n be real homogeneous functions in C ( n - { 0 } ) of degree k 2 , let ϕ ( x ) = ( ϕ 1 ( x ) , , ϕ n ( x ) ) and let μ be the Borel measure on 2 n given by μ ( E ) = n χ E ( x , ϕ ( x ) ) | x | γ - n d x where d x denotes the Lebesgue measure on n and γ > 0 . Let T μ be the convolution operator T μ f ( x ) = ( μ * f ) ( x ) and let E μ = { ( 1 / p , 1 / q ) T μ p , q < , 1 p , q } . Assume that, for x 0 , the following two conditions hold: det ( d 2 ϕ ( x ) h ) vanishes only at h = 0 and det ( d ϕ ( x ) ) 0 . In this paper we show that if γ > n ( k + 1 ) / 3 then E μ is the empty set and if γ n ( k + 1 ) / 3 then E μ is the closed segment with endpoints D = 1 - γ n ( k + 1 ) , 1 - 2 γ n ( k + 1 ) and D ' = 2 γ n ( 1 + k ) , γ n ( 1 + k ) . Also, we give some examples.

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