Extension Operator

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Leonhard Frerick - One of the best experts on this subject based on the ideXlab platform.

  • Whitney Extension Operators with arbitrary loss of differentiability
    Journal of Mathematical Analysis and Applications, 2020
    Co-Authors: Leonhard Frerick, Enrique Jorda, Arne Jakobs, Jochen Wengenroth
    Abstract:

    Abstract For a compact set K ⊂ R d we characterize the existence of a linear Extension Operator E : E ( K ) → C ∞ ( R d ) for the space of Whitney jets E ( K ) with a certain loss of derivatives σ, that is, the Operator satisfies the following continuity estimates for all n ∈ N 0 and all F ∈ E ( K ) sup ⁡ { | ∂ α E ( F ) ( x ) : | α | ≤ n , x ∈ R d | } ≤ C n ‖ F ‖ σ ( n ) , where ‖ ⋅ ‖ σ ( n ) denotes the Whitney norm and the map σ : N 0 → N 0 is monotonically increasing with σ ( n ) ≥ n and σ ( 0 ) = 0 . From our main result it follows directly that if a compact set K admits an Extension Operator, then it is always possible to construct a second Extension Operator resembling the original Whitney Operators E n : E n ( K ) → C n ( R d ) where the evaluations of the jet occurring in the Taylor polynomials are approximated by measures.

  • Extension Operators for smooth functions on compact subsets of the reals
    Mathematische Zeitschrift, 2019
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    We introduce sufficient as well as necessary conditions for a compact set K such that there is a continuous linear Extension Operator from the space of restrictions $$C^\infty (K)=\{F|_K: F\in C^\infty (\mathbb {R})\}$$ C ∞ ( K ) = { F | K : F ∈ C ∞ ( R ) } to $$C^\infty (\mathbb {R})$$ C ∞ ( R ) . This allows us to deal with examples of the form $$K=\{a_n:n\in \mathbb {N}\}\cup \{0\}$$ K = { a n : n ∈ N } ∪ { 0 } for $$a_n\rightarrow 0$$ a n → 0 previously considered by Fefferman and Ricci as well as Vogt.

  • Extension Operators for smooth functions on compact subsets of the reals
    arXiv: Functional Analysis, 2016
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    We introduce sufficient as well as necessary conditions for a compact set $K$ such that there is a continuous linear Extension Operator from the space of restrictions $C^\infty(K)=\lbrace F|_K: F\in C^\infty(\mathbb R)\rbrace$ to $C^\infty(\mathbb R)$. This allows us to deal with examples of the form $K=\lbrace a_n:n\in\mathbb N\rbrace \cup \lbrace 0\rbrace$ for $a_n\to 0$ previously considered by Fefferman and Ricci as well as Vogt.

  • Whitney Extension Operators without loss of derivatives
    Revista Matemática Iberoamericana, 2016
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    For a compact set K⊆Rd we characterize the existence of a linear Extension Operator E:E(K)→C∞(Rd) for the space of Whitney jets E(K) without loss of derivatives, that is, it satisfies the best possible continuity estimates sup{|∂αE(f)(x)|:|α|≤n,x∈Rd}≤Cn∥f∥n, where ∥⋅∥n denotes the nn-th Whitney norm. The characterization is by a surprisingly simple purely geometric condition introduced by Jonsson, Sjogren, and Wallis: there is ϱ∈(0,1) such that, for every x0∈K and ϵ∈(0,1), there are dd points x1…,xd in K∩B(x0,ϵ) satisfying dist(xn+1,\rm affine hull{x0,…,xn})≥ϱϵ for all n∈{0,…,d−1}.

  • Whitney Extension Operators without loss of derivatives
    arXiv: Functional Analysis, 2013
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    For a compact set, we characterize the existence of a linear Extension Operator E for the space of Whitney jets without loss of derivatives, that is, E satisfies the best possible continuity estimates: The supremum of all partial derivatives up to order n of E(f) is less or equal than a constant times the n-th Whitney norm of f. The characterization is a surprisingly simple purely geometric condition telling in a way that at all its points, the set is big enough in all directions.

Jochen Wengenroth - One of the best experts on this subject based on the ideXlab platform.

  • Whitney Extension Operators with arbitrary loss of differentiability
    Journal of Mathematical Analysis and Applications, 2020
    Co-Authors: Leonhard Frerick, Enrique Jorda, Arne Jakobs, Jochen Wengenroth
    Abstract:

    Abstract For a compact set K ⊂ R d we characterize the existence of a linear Extension Operator E : E ( K ) → C ∞ ( R d ) for the space of Whitney jets E ( K ) with a certain loss of derivatives σ, that is, the Operator satisfies the following continuity estimates for all n ∈ N 0 and all F ∈ E ( K ) sup ⁡ { | ∂ α E ( F ) ( x ) : | α | ≤ n , x ∈ R d | } ≤ C n ‖ F ‖ σ ( n ) , where ‖ ⋅ ‖ σ ( n ) denotes the Whitney norm and the map σ : N 0 → N 0 is monotonically increasing with σ ( n ) ≥ n and σ ( 0 ) = 0 . From our main result it follows directly that if a compact set K admits an Extension Operator, then it is always possible to construct a second Extension Operator resembling the original Whitney Operators E n : E n ( K ) → C n ( R d ) where the evaluations of the jet occurring in the Taylor polynomials are approximated by measures.

  • Extension Operators for smooth functions on compact subsets of the reals
    Mathematische Zeitschrift, 2019
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    We introduce sufficient as well as necessary conditions for a compact set K such that there is a continuous linear Extension Operator from the space of restrictions $$C^\infty (K)=\{F|_K: F\in C^\infty (\mathbb {R})\}$$ C ∞ ( K ) = { F | K : F ∈ C ∞ ( R ) } to $$C^\infty (\mathbb {R})$$ C ∞ ( R ) . This allows us to deal with examples of the form $$K=\{a_n:n\in \mathbb {N}\}\cup \{0\}$$ K = { a n : n ∈ N } ∪ { 0 } for $$a_n\rightarrow 0$$ a n → 0 previously considered by Fefferman and Ricci as well as Vogt.

  • Extension Operators for smooth functions on compact subsets of the reals
    arXiv: Functional Analysis, 2016
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    We introduce sufficient as well as necessary conditions for a compact set $K$ such that there is a continuous linear Extension Operator from the space of restrictions $C^\infty(K)=\lbrace F|_K: F\in C^\infty(\mathbb R)\rbrace$ to $C^\infty(\mathbb R)$. This allows us to deal with examples of the form $K=\lbrace a_n:n\in\mathbb N\rbrace \cup \lbrace 0\rbrace$ for $a_n\to 0$ previously considered by Fefferman and Ricci as well as Vogt.

  • Whitney Extension Operators without loss of derivatives
    Revista Matemática Iberoamericana, 2016
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    For a compact set K⊆Rd we characterize the existence of a linear Extension Operator E:E(K)→C∞(Rd) for the space of Whitney jets E(K) without loss of derivatives, that is, it satisfies the best possible continuity estimates sup{|∂αE(f)(x)|:|α|≤n,x∈Rd}≤Cn∥f∥n, where ∥⋅∥n denotes the nn-th Whitney norm. The characterization is by a surprisingly simple purely geometric condition introduced by Jonsson, Sjogren, and Wallis: there is ϱ∈(0,1) such that, for every x0∈K and ϵ∈(0,1), there are dd points x1…,xd in K∩B(x0,ϵ) satisfying dist(xn+1,\rm affine hull{x0,…,xn})≥ϱϵ for all n∈{0,…,d−1}.

  • Whitney Extension Operators without loss of derivatives
    arXiv: Functional Analysis, 2013
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    For a compact set, we characterize the existence of a linear Extension Operator E for the space of Whitney jets without loss of derivatives, that is, E satisfies the best possible continuity estimates: The supremum of all partial derivatives up to order n of E(f) is less or equal than a constant times the n-th Whitney norm of f. The characterization is a surprisingly simple purely geometric condition telling in a way that at all its points, the set is big enough in all directions.

Gabriela Kohr - One of the best experts on this subject based on the ideXlab platform.

  • g -Loewner chains, Bloch functions and Extension Operators in complex Banach spaces
    Analysis and Mathematical Physics, 2020
    Co-Authors: Ian D. Graham, Gabriela Kohr, Hidetaka Hamada, Mirela Kohr
    Abstract:

    Let Y be a complex Banach space and let $$r\ge 1$$. In this paper, we are concerned with an Extension Operator $$\varPhi _{\alpha , \beta }$$ that provides a way of extending a locally univalent function f on the unit disc $$\mathbb {U}$$ to a locally biholomorphic mapping $$F\in H(\varOmega _r)$$, where $$\varOmega _r=\{(z_1,w)\in \mathbb {C}\times Y: |z_1|^2+\Vert w\Vert _Y^r 0$$, $$\zeta \in \mathbb {U}$$, then $$F =\varPhi _{\alpha , \beta }(f)$$ can be embedded as the first element of a g-Loewner chain on $$\varOmega _r$$, for $$\alpha \in [0, 1]$$, $$\beta \in [0, 1/r]$$, $$\alpha +\beta \le 1$$. We also show that normalized univalent Bloch functions on $$\mathbb {U}$$ (resp. normalized uniformly locally univalent Bloch functions on $$\mathbb {U}$$) are extended to Bloch mappings on $$\varOmega _r$$ by $$\varPhi _{\alpha ,\beta }$$, for $$\alpha >0$$ and $$\beta \in [0,1/r)$$ (resp. for $$\alpha =0$$ and $$\beta \in [0,1/r]$$). In the case of the Muir type Extension Operator $$\varPhi _{P_k}$$, where $$k\ge 2$$ is an integer and $$P_k:Y\rightarrow \mathbb {C}$$ is a homogeneous polynomial mapping of degree k with $$\Vert P_k\Vert \le d(1,\partial g(\mathbb {U}))/4$$, we prove a similar Extension result for the first elements of g-Loewner chains on $$\varOmega _k$$. Next, we consider a modification of the Muir type Extension Operator $$\varPhi _{G,k}$$, where $$k\ge 2$$ is an integer and $$G:Y\rightarrow \mathbb {C}$$ is a holomorphic function such that $$G(0)=0$$ and $$DG(0)=0$$, and prove that if g is a univalent function with real coefficients on $$\mathbb {U}$$ such that $$g(0)=1$$, $$\mathfrak {R}g(\zeta )>0$$, $$\zeta \in \mathbb {U}$$, and g satisfies a natural boundary condition, and if the Extension Operator $$\varPhi _{G,k}$$ maps g-starlike functions from the unit disc $$\mathbb {U}$$ into starlike mappings on $$\varOmega _k$$, then G must be a homogeneous polynomial of degree at most k. We also obtain a preservation result of normalized uniformly locally univalent Bloch functions on $$\mathbb {U}$$ to Bloch mappings on $$\varOmega _k$$ by $$\varPhi _{P_k}$$.

  • Loewner chains, Bloch mappings and Pfaltzgraff-Suffridge Extension Operators on bounded symmetric domains
    Complex Variables and Elliptic Equations, 2019
    Co-Authors: Ian D. Graham, Hidetaka Hamada, Gabriela Kohr
    Abstract:

    ABSTRACTLet Y be a complex Banach space and let BY be the open unit ball of Y. In this paper we consider a generalization of the Pfaltzgraff-Suffridge Extension Operator on bounded symmetric domain...

  • LOEWNER CHAINS AND A MODIFICATION OF THE ROPER-SUFFRIDGE Extension Operator
    2007
    Co-Authors: Gabriela Kohr
    Abstract:

    In this paper we continue the study of the Roper-Suffridge Extension Operator. Let f be a locally univalent function on the unit disc and let Q : Cn−1 → C be a homogeneous polynomial of degree 2. We consider the family of Operators extending f to a holomorphic mapping from the unit ball B in C into C given by Φn,Q(f)(z) = (f(z1) +Q(z)f (z1), z(f (z1))), where z = (z2, . . . , zn). This Operator was recently introduced by Muir. In the case Q ≡ 0, this Operator reduces to the well known Roper-Suffridge Extension Operator. We prove that if f ∈ S then Φn,Q(f) ∈ S(B) whenever ‖Q‖ ≤ 1/4. Our proof yields Muir’s result that if f ∈ S∗ then Φn,Q(f) is also starlike on B. Moreover, if f ∈ K is imbedded in a convex subordination chain f(z1, t) over [0,∞) then Φn,Q(f) is also imbedded in a c.s.c. over [0,∞) on B whenever ‖Q‖ ≤ 1/2. MSC 2000. Primary: 32H. Secondary: 30C45.

  • The Roper-Suffridge Extension Operator and classes of biholomorphic mappings
    Science in China Series A: Mathematics, 2006
    Co-Authors: Ian D. Graham, Gabriela Kohr
    Abstract:

    In this paper we give a survey about the Roper-Suffridge Extension Operator and the developments in the theory of univalent mappings in several variables to which it has led. We begin with the basic geometric properties (most of which now have a number of different proofs) and discuss relations with the theory of Loewner chains and generalizations and modifications of the Operator, some of which are very recent.

  • Parametric representation and Extension Operators for biholomorphic mappings on some Reinhardt domains
    Complex Variables Theory and Application: An International Journal, 2005
    Co-Authors: Hidetaka Hamada, Gabriela Kohr, Mirela Kohr
    Abstract:

    In this article we obtain a generalization of the Pfaltzgraff–Suffridge Extension Operator, denoted by Φ n,p , on some Reinhardt domains in C n . We prove that if f∈ S 0(Bn ) and p≥ 2n/(n+1) then where . In particular, if f is starlike on the Euclidean unit ball B n then is starlike on Ω n,p . We also give examples of starlike mappings on Ω n,p . Moreover, we study certain convexity properties associated with the Operator Φ n,p . Further, we prove that the Roper–Suffridge Extension Operator Ψn preserves starlikeness of order 1/2. In the last section we obtain certain subordination results associated with the Operator Φ n,p .

Ian D. Graham - One of the best experts on this subject based on the ideXlab platform.

  • g -Loewner chains, Bloch functions and Extension Operators in complex Banach spaces
    Analysis and Mathematical Physics, 2020
    Co-Authors: Ian D. Graham, Gabriela Kohr, Hidetaka Hamada, Mirela Kohr
    Abstract:

    Let Y be a complex Banach space and let $$r\ge 1$$. In this paper, we are concerned with an Extension Operator $$\varPhi _{\alpha , \beta }$$ that provides a way of extending a locally univalent function f on the unit disc $$\mathbb {U}$$ to a locally biholomorphic mapping $$F\in H(\varOmega _r)$$, where $$\varOmega _r=\{(z_1,w)\in \mathbb {C}\times Y: |z_1|^2+\Vert w\Vert _Y^r 0$$, $$\zeta \in \mathbb {U}$$, then $$F =\varPhi _{\alpha , \beta }(f)$$ can be embedded as the first element of a g-Loewner chain on $$\varOmega _r$$, for $$\alpha \in [0, 1]$$, $$\beta \in [0, 1/r]$$, $$\alpha +\beta \le 1$$. We also show that normalized univalent Bloch functions on $$\mathbb {U}$$ (resp. normalized uniformly locally univalent Bloch functions on $$\mathbb {U}$$) are extended to Bloch mappings on $$\varOmega _r$$ by $$\varPhi _{\alpha ,\beta }$$, for $$\alpha >0$$ and $$\beta \in [0,1/r)$$ (resp. for $$\alpha =0$$ and $$\beta \in [0,1/r]$$). In the case of the Muir type Extension Operator $$\varPhi _{P_k}$$, where $$k\ge 2$$ is an integer and $$P_k:Y\rightarrow \mathbb {C}$$ is a homogeneous polynomial mapping of degree k with $$\Vert P_k\Vert \le d(1,\partial g(\mathbb {U}))/4$$, we prove a similar Extension result for the first elements of g-Loewner chains on $$\varOmega _k$$. Next, we consider a modification of the Muir type Extension Operator $$\varPhi _{G,k}$$, where $$k\ge 2$$ is an integer and $$G:Y\rightarrow \mathbb {C}$$ is a holomorphic function such that $$G(0)=0$$ and $$DG(0)=0$$, and prove that if g is a univalent function with real coefficients on $$\mathbb {U}$$ such that $$g(0)=1$$, $$\mathfrak {R}g(\zeta )>0$$, $$\zeta \in \mathbb {U}$$, and g satisfies a natural boundary condition, and if the Extension Operator $$\varPhi _{G,k}$$ maps g-starlike functions from the unit disc $$\mathbb {U}$$ into starlike mappings on $$\varOmega _k$$, then G must be a homogeneous polynomial of degree at most k. We also obtain a preservation result of normalized uniformly locally univalent Bloch functions on $$\mathbb {U}$$ to Bloch mappings on $$\varOmega _k$$ by $$\varPhi _{P_k}$$.

  • Loewner chains, Bloch mappings and Pfaltzgraff-Suffridge Extension Operators on bounded symmetric domains
    Complex Variables and Elliptic Equations, 2019
    Co-Authors: Ian D. Graham, Hidetaka Hamada, Gabriela Kohr
    Abstract:

    ABSTRACTLet Y be a complex Banach space and let BY be the open unit ball of Y. In this paper we consider a generalization of the Pfaltzgraff-Suffridge Extension Operator on bounded symmetric domain...

  • The Roper-Suffridge Extension Operator and classes of biholomorphic mappings
    Science in China Series A: Mathematics, 2006
    Co-Authors: Ian D. Graham, Gabriela Kohr
    Abstract:

    In this paper we give a survey about the Roper-Suffridge Extension Operator and the developments in the theory of univalent mappings in several variables to which it has led. We begin with the basic geometric properties (most of which now have a number of different proofs) and discuss relations with the theory of Loewner chains and generalizations and modifications of the Operator, some of which are very recent.

  • Geometric Function Theory in One and Higher Dimensions
    2003
    Co-Authors: Ian D. Graham, Gabriela Kohr
    Abstract:

    Univalent functions: elementary properties of univalent functions Subclasses of univalent functions in the unit disc The Loewner theory Bloch functions and the Bloch constant Linear invariance in the unit disc Univalent mappings in several complex variables and complex Banach spaces Univalence in several complex variables Growth, covering and distortion results for starlike and convex mappings in Cn and complex Banach spaces Loewner chains in several complex variables Bloch constant problems in several complex variables Linear invariance in several complex variables Univalent mappings and the Roper-Suffridge Extension Operator.

  • loewner chains and the roper suffridge Extension Operator
    Journal of Mathematical Analysis and Applications, 2000
    Co-Authors: Ian D. Graham, Gabriela Kohr, Mirela Kohr
    Abstract:

    Abstract Let f be a locally univalent function on the unit disc and let α ∈ [0,  1 2 ]. We consider the family of Operators extending f to a holomorphic map from the unit ball B in Cn to Cn given by Φn, α(f)(z) = (f(z1), z′(f′(z1))α), where z′ = (z2,…,zn). When α =  1 2 we obtain the Roper–Suffridge Extension Operator. We show that if f ∈ S then Φn, α(f) can be imbedded in a Loewner chain. Our proof shows that if f ∈ S* then Φn, α(f) is starlike, and if f ∈ Ŝβ with |β| π 2 then Φn, α(f) is a spirallike map of type β. In particular we obtain a new proof that the Roper–Suffridge Operator preserves starlikeness. We also obtain the radius of starlikeness of Φn, α(S) and the radius of convexity of Φn, 1/2(S). We show that if f is a normalized univalent Bloch function on U then Φn, α(f) is a Bloch mapping on B. Finally we show that if f belongs to a class of univalent functions which satisfy growth and distortion results, then Φn, α(f) satisfies related growth and covering results.

Enrique Jorda - One of the best experts on this subject based on the ideXlab platform.

  • Whitney Extension Operators with arbitrary loss of differentiability
    Journal of Mathematical Analysis and Applications, 2020
    Co-Authors: Leonhard Frerick, Enrique Jorda, Arne Jakobs, Jochen Wengenroth
    Abstract:

    Abstract For a compact set K ⊂ R d we characterize the existence of a linear Extension Operator E : E ( K ) → C ∞ ( R d ) for the space of Whitney jets E ( K ) with a certain loss of derivatives σ, that is, the Operator satisfies the following continuity estimates for all n ∈ N 0 and all F ∈ E ( K ) sup ⁡ { | ∂ α E ( F ) ( x ) : | α | ≤ n , x ∈ R d | } ≤ C n ‖ F ‖ σ ( n ) , where ‖ ⋅ ‖ σ ( n ) denotes the Whitney norm and the map σ : N 0 → N 0 is monotonically increasing with σ ( n ) ≥ n and σ ( 0 ) = 0 . From our main result it follows directly that if a compact set K admits an Extension Operator, then it is always possible to construct a second Extension Operator resembling the original Whitney Operators E n : E n ( K ) → C n ( R d ) where the evaluations of the jet occurring in the Taylor polynomials are approximated by measures.

  • Extension Operators for smooth functions on compact subsets of the reals
    Mathematische Zeitschrift, 2019
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    We introduce sufficient as well as necessary conditions for a compact set K such that there is a continuous linear Extension Operator from the space of restrictions $$C^\infty (K)=\{F|_K: F\in C^\infty (\mathbb {R})\}$$ C ∞ ( K ) = { F | K : F ∈ C ∞ ( R ) } to $$C^\infty (\mathbb {R})$$ C ∞ ( R ) . This allows us to deal with examples of the form $$K=\{a_n:n\in \mathbb {N}\}\cup \{0\}$$ K = { a n : n ∈ N } ∪ { 0 } for $$a_n\rightarrow 0$$ a n → 0 previously considered by Fefferman and Ricci as well as Vogt.

  • Extension Operators for smooth functions on compact subsets of the reals
    arXiv: Functional Analysis, 2016
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    We introduce sufficient as well as necessary conditions for a compact set $K$ such that there is a continuous linear Extension Operator from the space of restrictions $C^\infty(K)=\lbrace F|_K: F\in C^\infty(\mathbb R)\rbrace$ to $C^\infty(\mathbb R)$. This allows us to deal with examples of the form $K=\lbrace a_n:n\in\mathbb N\rbrace \cup \lbrace 0\rbrace$ for $a_n\to 0$ previously considered by Fefferman and Ricci as well as Vogt.

  • Whitney Extension Operators without loss of derivatives
    Revista Matemática Iberoamericana, 2016
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    For a compact set K⊆Rd we characterize the existence of a linear Extension Operator E:E(K)→C∞(Rd) for the space of Whitney jets E(K) without loss of derivatives, that is, it satisfies the best possible continuity estimates sup{|∂αE(f)(x)|:|α|≤n,x∈Rd}≤Cn∥f∥n, where ∥⋅∥n denotes the nn-th Whitney norm. The characterization is by a surprisingly simple purely geometric condition introduced by Jonsson, Sjogren, and Wallis: there is ϱ∈(0,1) such that, for every x0∈K and ϵ∈(0,1), there are dd points x1…,xd in K∩B(x0,ϵ) satisfying dist(xn+1,\rm affine hull{x0,…,xn})≥ϱϵ for all n∈{0,…,d−1}.

  • Whitney Extension Operators without loss of derivatives
    arXiv: Functional Analysis, 2013
    Co-Authors: Leonhard Frerick, Enrique Jorda, Jochen Wengenroth
    Abstract:

    For a compact set, we characterize the existence of a linear Extension Operator E for the space of Whitney jets without loss of derivatives, that is, E satisfies the best possible continuity estimates: The supremum of all partial derivatives up to order n of E(f) is less or equal than a constant times the n-th Whitney norm of f. The characterization is a surprisingly simple purely geometric condition telling in a way that at all its points, the set is big enough in all directions.

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