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January 21, 2011

Harmonic Analysis Lecture Notes 8

Filed under: 2011 Harmonic Analysis Course — xuhmath @ 6:59 pm

Note that

  1. {|f| \geq |g|} then {||f||_{L^{p,q}}^* \geq ||g||_{L^{p,q}}^*}.
  2. The quasi-norms are rearrangement invariant.

Proposition If {f \in L^{p,q}}, then write {f = \sum f_m} where

\displaystyle  	f_m(x) = f(x) \chi_{\{x : 2^m \leq |f(x)| < 2^{m+1}\}}.

Then

\displaystyle  	||f||_{L_{p,q}^*} \approx_{p,q} \big|\big| \; ||f_m||_{L^p({\mathbb R}^d)} \big|\big|_{\ell^q_{m}({\mathbb Z})}

In partiular, {L^{p,q_1} \subseteq L^{p,q_2}} whenever {q_1 \leq q_2}.

Proof: It suffices to consider {f} of the form {f = \sum 2^m \chi_{E_m}} with disjoint sets {E_m} (cf. {E_m = \{ 2^m \leq |f| < 2^{m+1}\}.})

\displaystyle  \begin{array}{rcl}  	\left( ||f||^*_{L^{p,q}}\right)^q &=& \displaystyle p \int_0^\infty \lambda^q |\{ |f| > \lambda \} |^{q/p} \frac{d\lambda}{\lambda} \\ \\ 	&\approx& \displaystyle \sum_m \int_{2^{m-1}}^{2^m} \lambda^q \left(\sum_{n\geq m} |E_n|\right)^{q/p} \frac{d\lambda}{\lambda} \\ \\ 	&\approx& \displaystyle \sum_m \left[ 2^m \left( \sum_{n\geq m} |E_n| \right)^{1/p} \right]^q 	\end{array}

Thus

\displaystyle  	\sum_m \big| 2^m |E_m|^{1/p} \big|^q \underbrace{\lesssim}_{\text{keep only }n=m} (||f||_{L_{p,q}}^*)^q \underbrace{\lesssim}_{\textit{concavity of fractional powers}} \sum_{m \leq n} \big| 2^m |E_n|^{1/p} \big|^q.

and so

\displaystyle  	\sum_m ||f_m||_{L^p}^q \lesssim ( ||f||_{L^{p,q}}^*)^q \lesssim \sum_{m \leq n} 2^{-q(n-m)} [ 2^n |E_n|^{1/p}]^q \lesssim (1-2^{-q})^{-1} \big|\big| ||f_n||_{L^p} \big|\big|_{\ell^q_n({\mathbb Z})}

\Box

Things to note: Make continuum problem discrete, concavity of fractional powers

Proposition For {1 < p \leq \infty}

  1. {||f||_{L_{p,q}^*} \approx \sup \{ |\int fg| : ||g||_{L^{p',q'}}^* \leq 1\}}
  2. The right hand side of part (1) defines an actual norm.
  3. {(L^{p,q})^* \cong L^{p',q'}} as Banach spaces, provided {q \neq \infty}

Why not {p=1}? Proof: For assertion (1)

\displaystyle  \begin{array}{rcl}  	\bigg| \int fg \bigg| &\leq& \displaystyle \int_0^\infty \int_0^\infty \int_{{\mathbb R}^d} \chi_{\{|f| > \lambda\}} \chi_{\{|g| > \mu\}} \; dx \; d\mu \; d\lambda \\ \\ 	&\lesssim& \displaystyle \int_0^\infty \int_0^\infty | \{ |f| > \lambda\} \cap \{ |g| > \mu \} | \; d\lambda \; d\mu 	 	\end{array}

and a miracle occurs.

For assertion (3), it suffices to consider {f = \sum 2^m \chi_{E_m}}, which {E_m} disjoint, and of {L^{p,q}} “norm” 1. Choosing

\displaystyle  	g = \sum_m \left( 2^m |E_m|^\frac{1}{p} \right)^{q-1} |E_m|^{-\frac{1}{p'}} \chi_{E_m}

then

\displaystyle  \begin{array}{rcl}  	\int fg &=& \displaystyle \sum_m \left( 2^m |E_m|^\frac{1}{p} \right)^{q-1} 2^{m} |E_m|^{1-\frac{1}{p'}} \\ \\ 	&=& \displaystyle \sum_m \left( 2^m |E_m|^\frac{1}{p}\right)^{q-1} 2^m |E_m|^\frac{1}{p} \\ \\ 	&\approx& ||f||_{L^{p,q}}^* \approx 1 	\end{array}

and

\displaystyle  \begin{array}{rcl}  	||g||_{L^{p',q'}}^* \approx \ldots \text{stay tuned} 	\end{array}

\Box

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