Integrating elements of Holder-Besov spaces using LIttlewood-Payley decomposition
$begingroup$
Let $ mathcal{C}^{s} $ be the Holder-Besov space equipped with the norm
$$ | u |_{mathcal{C}^{s}} = sup_{j geq -1}Big| 2^{js} | mathit{Delta_{j}u} |_{L^{infty}} Big|.$$
Suppose that that $ s $ is negative. I have shown that for every test function $ phi in mathcal{D} $ supported in $ [-T,T] $. We have
$$bigg| phi sum_{j geq -1} int_{0}^{t} Delta_{j}u (t')dt' bigg| _{mathcal{C}^{s+1}} leq (1+T)| phi |_{C^{1}_{b}} |u |_{mathcal{C}^{s}}.$$
How do I show that if $u$ have compact support. Then
$$bigg| sum_{j geq -1} int_{0}^{t} Delta_{j}u (t')dt' bigg| _{mathcal{C}^{s+1}} leq (1+T) |u |_{mathcal{C}^{s}}.$$
It is easy to see how to make $| phi |_{C^{1}_{b}}$ vanish on the RHS. But I am not sure how to do it on the LFH.
holder-spaces besov-space
asked Dec 4 '18 at 9:50
MeagainMeagain
16910
$endgroup$
add a comment |
$begingroup$
Let $ mathcal{C}^{s} $ be the Holder-Besov space equipped with the norm
$$ | u |_{mathcal{C}^{s}} = sup_{j geq -1}Big| 2^{js} | mathit{Delta_{j}u} |_{L^{infty}} Big|.$$
Suppose that that $ s $ is negative. I have shown that for every test function $ phi in mathcal{D} $ supported in $ [-T,T] $. We have
$$bigg| phi sum_{j geq -1} int_{0}^{t} Delta_{j}u (t')dt' bigg| _{mathcal{C}^{s+1}} leq (1+T)| phi |_{C^{1}_{b}} |u |_{mathcal{C}^{s}}.$$
How do I show that if $u$ have compact support. Then
$$bigg| sum_{j geq -1} int_{0}^{t} Delta_{j}u (t')dt' bigg| _{mathcal{C}^{s+1}} leq (1+T) |u |_{mathcal{C}^{s}}.$$
It is easy to see how to make $| phi |_{C^{1}_{b}}$ vanish on the RHS. But I am not sure how to do it on the LFH.
holder-spaces besov-space
asked Dec 4 '18 at 9:50
MeagainMeagain
16910
$endgroup$
add a comment |
$begingroup$
Let $ mathcal{C}^{s} $ be the Holder-Besov space equipped with the norm
$$ | u |_{mathcal{C}^{s}} = sup_{j geq -1}Big| 2^{js} | mathit{Delta_{j}u} |_{L^{infty}} Big|.$$
Suppose that that $ s $ is negative. I have shown that for every test function $ phi in mathcal{D} $ supported in $ [-T,T] $. We have
$$bigg| phi sum_{j geq -1} int_{0}^{t} Delta_{j}u (t')dt' bigg| _{mathcal{C}^{s+1}} leq (1+T)| phi |_{C^{1}_{b}} |u |_{mathcal{C}^{s}}.$$
How do I show that if $u$ have compact support. Then
$$bigg| sum_{j geq -1} int_{0}^{t} Delta_{j}u (t')dt' bigg| _{mathcal{C}^{s+1}} leq (1+T) |u |_{mathcal{C}^{s}}.$$
It is easy to see how to make $| phi |_{C^{1}_{b}}$ vanish on the RHS. But I am not sure how to do it on the LFH.
holder-spaces besov-space
asked Dec 4 '18 at 9:50
MeagainMeagain
16910
$endgroup$
Let $ mathcal{C}^{s} $ be the Holder-Besov space equipped with the norm
$$ | u |_{mathcal{C}^{s}} = sup_{j geq -1}Big| 2^{js} | mathit{Delta_{j}u} |_{L^{infty}} Big|.$$
Suppose that that $ s $ is negative. I have shown that for every test function $ phi in mathcal{D} $ supported in $ [-T,T] $. We have
$$bigg| phi sum_{j geq -1} int_{0}^{t} Delta_{j}u (t')dt' bigg| _{mathcal{C}^{s+1}} leq (1+T)| phi |_{C^{1}_{b}} |u |_{mathcal{C}^{s}}.$$
How do I show that if $u$ have compact support. Then
$$bigg| sum_{j geq -1} int_{0}^{t} Delta_{j}u (t')dt' bigg| _{mathcal{C}^{s+1}} leq (1+T) |u |_{mathcal{C}^{s}}.$$
It is easy to see how to make $| phi |_{C^{1}_{b}}$ vanish on the RHS. But I am not sure how to do it on the LFH.
holder-spaces besov-space
holder-spaces besov-space
asked Dec 4 '18 at 9:50
MeagainMeagain
16910
asked Dec 4 '18 at 9:50
MeagainMeagain
16910
asked Dec 4 '18 at 9:50
MeagainMeagain
16910
asked Dec 4 '18 at 9:50
MeagainMeagain
16910
asked Dec 4 '18 at 9:50
MeagainMeagain
16910
16910
add a comment |
add a comment |
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