Different definitions of Besov norm/space The 2019 Stack Overflow Developer Survey Results Are InModuli of smoothness, Besov spaces, and Sobolev spacesBesov spaces---concrete description of spatial inhomogeneityDifferentiation in Besov–Zygmund spacesReference request: The compactness and compact embedding in Besov Space?Simple Inequality for Proving Equivalent Besov SeminormsRelation between Besov and Sobolev spaces (Littlewood-Paley-theory)Sobolev space definitions$mathcalC^alpha$ Besov spaces: DefinitionEmbedding property of homogeneous periodic Besov spacesHow to prove that a delta function belongs to the Besov space $B^-1$?
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Different definitions of Besov norm/space
The 2019 Stack Overflow Developer Survey Results Are InModuli of smoothness, Besov spaces, and Sobolev spacesBesov spaces---concrete description of spatial inhomogeneityDifferentiation in Besov–Zygmund spacesReference request: The compactness and compact embedding in Besov Space?Simple Inequality for Proving Equivalent Besov SeminormsRelation between Besov and Sobolev spaces (Littlewood-Paley-theory)Sobolev space definitions$mathcalC^alpha$ Besov spaces: DefinitionEmbedding property of homogeneous periodic Besov spacesHow to prove that a delta function belongs to the Besov space $B^-1$?
$begingroup$
I'm following two "different" approaches to the Besov Spaces, but I don't get if the two definitions given are equivalent.
Victor I. Burenkov - Sobolev Spaces On Domains.
Given $f:mathbbR^n to mathbbR$ and $h in mathbbR^n$ we set $Delta_h f(x) = f(x+h)-f(x)$ and recursively $Delta_h ^sigma f(x) = underbraceDelta_h (Delta_h(dots (Delta_h_sigma text times f )dots))(x) = sum_k=0^sigma (-1)^sigma binomsigmak f(x+kh) $.
Definition. Let $l>0, sigma in mathbbN, sigma > l, 1 le p, theta le infty$. The function $f$ belongs to the Besov-Nikol'skij space $B^l _p,theta (mathbbR^n)$ if $f$ is measurable on $mathbbR^n$ and $$|f|_B^l _p,theta (mathbbR^n) = |f|_L^p (mathbbR^n) + |f|_b^l _p,theta (mathbbR^n)<infty$$ where $$|f|_b^l _p,theta (mathbbR^n) = left( int_mathbbR^n left(frac_L^p(mathbbR^n)^l right)^theta fracdhh right)^1/theta$$if $1 le theta < infty$ and $$|f|_B^l _p,infty (mathbbR^n) = sup_h in mathbbR^n setminus 0 frac_L^p(mathbbR^n)^l. $$
Burenkov says also that the definition is independent of $sigma > l$.
Giovanni Leoni - A First Course in Sobolev Spaces.
Given a function $u: mathbbR^N to mathbbR$, for every $h in mathbbR$, $i=1, dots, N$, and $x in mathbbR^n$, we define $$Delta^h _i u(x) := u(x+h e_i)-u(x)$$ where $e_i$ is the ith vector of the canonical basis in $mathbbR^n$.
Definition. Let $1 le p, theta le infty$ and $0<s<1$. A function $u in L^1_textloc (mathbbR^N)$ belongs to the Besov space $B^s,p,theta(mathbbR^N)$ if $$|u|_B^s,p,theta (mathbbR^N) = |u|_L^p (mathbbR^N) + |u|_B^s,p,theta (mathbbR^N)<infty$$ where $$|u|_B^s,p,theta (mathbbR^N) := sum_i=1^N left( int_0^infty | Delta^h _i u |^theta _L^p(mathbbR^N) fracdhh^1 +stheta right)^1/theta$$if $theta < infty$ and $$|u|_B^s,p,infty(mathbbR^N) := sum_i=1^N sup_h>0 frac1h^s | Delta_i ^h u |_L^p(mathbbR^N).$$
The two definition look like pretty the same but, at least apparently, they aren't. Are they actually equivalent? I don't need a proof, but I just wanted to know if this is a field of Math in which things with the same name are not equivalently defined (Burenkov defines other nine equivalent Besov norms, but there's not the Leoni's one between them).
Thanks in advance!
functional-analysis sobolev-spaces besov-space
asked Jul 19 '16 at 9:36
gangrenegangrene
915514
$endgroup$
add a comment |
$begingroup$
I'm following two "different" approaches to the Besov Spaces, but I don't get if the two definitions given are equivalent.
Victor I. Burenkov - Sobolev Spaces On Domains.
Given $f:mathbbR^n to mathbbR$ and $h in mathbbR^n$ we set $Delta_h f(x) = f(x+h)-f(x)$ and recursively $Delta_h ^sigma f(x) = underbraceDelta_h (Delta_h(dots (Delta_h_sigma text times f )dots))(x) = sum_k=0^sigma (-1)^sigma binomsigmak f(x+kh) $.
Definition. Let $l>0, sigma in mathbbN, sigma > l, 1 le p, theta le infty$. The function $f$ belongs to the Besov-Nikol'skij space $B^l _p,theta (mathbbR^n)$ if $f$ is measurable on $mathbbR^n$ and $$|f|_B^l _p,theta (mathbbR^n) = |f|_L^p (mathbbR^n) + |f|_b^l _p,theta (mathbbR^n)<infty$$ where $$|f|_b^l _p,theta (mathbbR^n) = left( int_mathbbR^n left(frac_L^p(mathbbR^n)^l right)^theta fracdhh right)^1/theta$$if $1 le theta < infty$ and $$|f|_B^l _p,infty (mathbbR^n) = sup_h in mathbbR^n setminus 0 frac_L^p(mathbbR^n)^l. $$
Burenkov says also that the definition is independent of $sigma > l$.
Giovanni Leoni - A First Course in Sobolev Spaces.
Given a function $u: mathbbR^N to mathbbR$, for every $h in mathbbR$, $i=1, dots, N$, and $x in mathbbR^n$, we define $$Delta^h _i u(x) := u(x+h e_i)-u(x)$$ where $e_i$ is the ith vector of the canonical basis in $mathbbR^n$.
Definition. Let $1 le p, theta le infty$ and $0<s<1$. A function $u in L^1_textloc (mathbbR^N)$ belongs to the Besov space $B^s,p,theta(mathbbR^N)$ if $$|u|_B^s,p,theta (mathbbR^N) = |u|_L^p (mathbbR^N) + |u|_B^s,p,theta (mathbbR^N)<infty$$ where $$|u|_B^s,p,theta (mathbbR^N) := sum_i=1^N left( int_0^infty | Delta^h _i u |^theta _L^p(mathbbR^N) fracdhh^1 +stheta right)^1/theta$$if $theta < infty$ and $$|u|_B^s,p,infty(mathbbR^N) := sum_i=1^N sup_h>0 frac1h^s | Delta_i ^h u |_L^p(mathbbR^N).$$
The two definition look like pretty the same but, at least apparently, they aren't. Are they actually equivalent? I don't need a proof, but I just wanted to know if this is a field of Math in which things with the same name are not equivalently defined (Burenkov defines other nine equivalent Besov norms, but there's not the Leoni's one between them).
Thanks in advance!
functional-analysis sobolev-spaces besov-space
asked Jul 19 '16 at 9:36
gangrenegangrene
915514
$endgroup$
add a comment |
$begingroup$
I'm following two "different" approaches to the Besov Spaces, but I don't get if the two definitions given are equivalent.
Victor I. Burenkov - Sobolev Spaces On Domains.
Given $f:mathbbR^n to mathbbR$ and $h in mathbbR^n$ we set $Delta_h f(x) = f(x+h)-f(x)$ and recursively $Delta_h ^sigma f(x) = underbraceDelta_h (Delta_h(dots (Delta_h_sigma text times f )dots))(x) = sum_k=0^sigma (-1)^sigma binomsigmak f(x+kh) $.
Definition. Let $l>0, sigma in mathbbN, sigma > l, 1 le p, theta le infty$. The function $f$ belongs to the Besov-Nikol'skij space $B^l _p,theta (mathbbR^n)$ if $f$ is measurable on $mathbbR^n$ and $$|f|_B^l _p,theta (mathbbR^n) = |f|_L^p (mathbbR^n) + |f|_b^l _p,theta (mathbbR^n)<infty$$ where $$|f|_b^l _p,theta (mathbbR^n) = left( int_mathbbR^n left(frac_L^p(mathbbR^n)^l right)^theta fracdhh right)^1/theta$$if $1 le theta < infty$ and $$|f|_B^l _p,infty (mathbbR^n) = sup_h in mathbbR^n setminus 0 frac_L^p(mathbbR^n)^l. $$
Burenkov says also that the definition is independent of $sigma > l$.
Giovanni Leoni - A First Course in Sobolev Spaces.
Given a function $u: mathbbR^N to mathbbR$, for every $h in mathbbR$, $i=1, dots, N$, and $x in mathbbR^n$, we define $$Delta^h _i u(x) := u(x+h e_i)-u(x)$$ where $e_i$ is the ith vector of the canonical basis in $mathbbR^n$.
Definition. Let $1 le p, theta le infty$ and $0<s<1$. A function $u in L^1_textloc (mathbbR^N)$ belongs to the Besov space $B^s,p,theta(mathbbR^N)$ if $$|u|_B^s,p,theta (mathbbR^N) = |u|_L^p (mathbbR^N) + |u|_B^s,p,theta (mathbbR^N)<infty$$ where $$|u|_B^s,p,theta (mathbbR^N) := sum_i=1^N left( int_0^infty | Delta^h _i u |^theta _L^p(mathbbR^N) fracdhh^1 +stheta right)^1/theta$$if $theta < infty$ and $$|u|_B^s,p,infty(mathbbR^N) := sum_i=1^N sup_h>0 frac1h^s | Delta_i ^h u |_L^p(mathbbR^N).$$
The two definition look like pretty the same but, at least apparently, they aren't. Are they actually equivalent? I don't need a proof, but I just wanted to know if this is a field of Math in which things with the same name are not equivalently defined (Burenkov defines other nine equivalent Besov norms, but there's not the Leoni's one between them).
Thanks in advance!
functional-analysis sobolev-spaces besov-space
asked Jul 19 '16 at 9:36
gangrenegangrene
915514
$endgroup$
I'm following two "different" approaches to the Besov Spaces, but I don't get if the two definitions given are equivalent.
Victor I. Burenkov - Sobolev Spaces On Domains.
Given $f:mathbbR^n to mathbbR$ and $h in mathbbR^n$ we set $Delta_h f(x) = f(x+h)-f(x)$ and recursively $Delta_h ^sigma f(x) = underbraceDelta_h (Delta_h(dots (Delta_h_sigma text times f )dots))(x) = sum_k=0^sigma (-1)^sigma binomsigmak f(x+kh) $.
Definition. Let $l>0, sigma in mathbbN, sigma > l, 1 le p, theta le infty$. The function $f$ belongs to the Besov-Nikol'skij space $B^l _p,theta (mathbbR^n)$ if $f$ is measurable on $mathbbR^n$ and $$|f|_B^l _p,theta (mathbbR^n) = |f|_L^p (mathbbR^n) + |f|_b^l _p,theta (mathbbR^n)<infty$$ where $$|f|_b^l _p,theta (mathbbR^n) = left( int_mathbbR^n left(frac_L^p(mathbbR^n)^l right)^theta fracdhh right)^1/theta$$if $1 le theta < infty$ and $$|f|_B^l _p,infty (mathbbR^n) = sup_h in mathbbR^n setminus 0 frac_L^p(mathbbR^n)^l. $$
Burenkov says also that the definition is independent of $sigma > l$.
Giovanni Leoni - A First Course in Sobolev Spaces.
Given a function $u: mathbbR^N to mathbbR$, for every $h in mathbbR$, $i=1, dots, N$, and $x in mathbbR^n$, we define $$Delta^h _i u(x) := u(x+h e_i)-u(x)$$ where $e_i$ is the ith vector of the canonical basis in $mathbbR^n$.
Definition. Let $1 le p, theta le infty$ and $0<s<1$. A function $u in L^1_textloc (mathbbR^N)$ belongs to the Besov space $B^s,p,theta(mathbbR^N)$ if $$|u|_B^s,p,theta (mathbbR^N) = |u|_L^p (mathbbR^N) + |u|_B^s,p,theta (mathbbR^N)<infty$$ where $$|u|_B^s,p,theta (mathbbR^N) := sum_i=1^N left( int_0^infty | Delta^h _i u |^theta _L^p(mathbbR^N) fracdhh^1 +stheta right)^1/theta$$if $theta < infty$ and $$|u|_B^s,p,infty(mathbbR^N) := sum_i=1^N sup_h>0 frac1h^s | Delta_i ^h u |_L^p(mathbbR^N).$$
The two definition look like pretty the same but, at least apparently, they aren't. Are they actually equivalent? I don't need a proof, but I just wanted to know if this is a field of Math in which things with the same name are not equivalently defined (Burenkov defines other nine equivalent Besov norms, but there's not the Leoni's one between them).
Thanks in advance!
functional-analysis sobolev-spaces besov-space
functional-analysis sobolev-spaces besov-space
asked Jul 19 '16 at 9:36
gangrenegangrene
915514
asked Jul 19 '16 at 9:36
gangrenegangrene
915514
asked Jul 19 '16 at 9:36
gangrenegangrene
915514
asked Jul 19 '16 at 9:36
gangrenegangrene
915514
asked Jul 19 '16 at 9:36
gangrenegangrene
915514
915514
add a comment |
add a comment |
1 Answer
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The answer is positive: these norms are equivalent (the second definition was given only for $0< s < 1$, but it can be extended in a natural way to any $s>0$).
edited Mar 30 at 12:03
Daniele Tampieri
2,66221022
answered Mar 30 at 10:58
V..I. BurenkovV..I. Burenkov
111
$endgroup$
add a comment |
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$begingroup$
The answer is positive: these norms are equivalent (the second definition was given only for $0< s < 1$, but it can be extended in a natural way to any $s>0$).
edited Mar 30 at 12:03
Daniele Tampieri
2,66221022
answered Mar 30 at 10:58
V..I. BurenkovV..I. Burenkov
111
$endgroup$
add a comment |
$begingroup$
The answer is positive: these norms are equivalent (the second definition was given only for $0< s < 1$, but it can be extended in a natural way to any $s>0$).
edited Mar 30 at 12:03
Daniele Tampieri
2,66221022
answered Mar 30 at 10:58
V..I. BurenkovV..I. Burenkov
111
$endgroup$
add a comment |
$begingroup$
The answer is positive: these norms are equivalent (the second definition was given only for $0< s < 1$, but it can be extended in a natural way to any $s>0$).
edited Mar 30 at 12:03
Daniele Tampieri
2,66221022
answered Mar 30 at 10:58
V..I. BurenkovV..I. Burenkov
111
$endgroup$
The answer is positive: these norms are equivalent (the second definition was given only for $0< s < 1$, but it can be extended in a natural way to any $s>0$).
edited Mar 30 at 12:03
Daniele Tampieri
2,66221022
answered Mar 30 at 10:58
V..I. BurenkovV..I. Burenkov
111
edited Mar 30 at 12:03
Daniele Tampieri
2,66221022
edited Mar 30 at 12:03
Daniele Tampieri
2,66221022
edited Mar 30 at 12:03
Daniele Tampieri
2,66221022
2,66221022
answered Mar 30 at 10:58
V..I. BurenkovV..I. Burenkov
111
answered Mar 30 at 10:58
V..I. BurenkovV..I. Burenkov
111
answered Mar 30 at 10:58
V..I. BurenkovV..I. Burenkov
111
111
add a comment |
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