Spline and Ciesielski Systems

In the first four sections we consider the Kronecker product of the unbounded Ciesielski or spline systems % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWGObWaa0baaSqaaiaad6gaaeaacaGG % OaGaamyBamaaBaaameaacaWGQbaabeaaliaacYcacaWGRbWaaSbaaW % qaaiaadQgaaeqaaSGaaiykaaaaaaa!4765! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ h_n^{({m_j},{k_j})} $$ of order (m _j , k _j ) introduced by Ciesielski [38, 45, 46] % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaGGOaWaaqWaaeaacaWGRbWaaSbaaSqa % aiaadQgaaeqaaaGccaGLhWUaayjcSdGaeyizImQaamyBamaaBaaale % aacaWGQbaabeaakiabgUcaRiaaigdacaGGSaGaamyBamaaBaaaleaa % caWGQbaabeaakiabgwMiZkabgkHiTiaaigdacaGGPaaaaa!514E! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ (\left| {{k_j}} \right| \leqslant {m_j} + 1,{m_j} \geqslant - 1) $$ and the Hardy spaces % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWGibWaaSbaaSqaaiaadcfacaGGSaGa % amyuaaqabaGccqGH9aqpcaWGibWaa0baaSqaaiaadchacaGGSaGaam % yCaaqaaiaacIcacaWGTbGaaiilaiaadUgacaGGPaaaaaaa!4B04! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ {H_{P,Q}} = H_{p,q}^{(m,k)} $$ and % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWGibWaa0baaSqaaiaadcfacaGGSaGa % amyuaaqaaiablgAjxbaakiabg2da9iaadIeadaqhaaWcbaGaamiCai % aacYcacaWGXbaabaGaeSyOLCLaaiikaiaad2gacaGGSaGaam4Aaiaa % cMcaaaaaaa!4E61! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ H_{P,Q}^\square = H_{p,q}^{\square (m,k)} $$ defined by the L _p,q norms of the non-tangential maximal functions. Note that if k _j = m _j + 1 for all j then we obtain the usual dyadic Hardy spaces, in other cases H _p,q is the restriction of the classical H _p,q space to the unit cube. It is known that % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWGObWaa0baaSqaaiaad6gaaeaacaGG % OaGaamyBamaaBaaameaacaWGQbaabeaaliaacYcacaWGRbWaaSbaaW % qaaiaadQgaaeqaaSGaaiykaaaaaaa!4765! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ h_n^{({m_j},{k_j})} $$ and % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaGGOaGaamiAaiaad2gadaqhaaWcbaGa % amOBaaqaaiaacIcacaWGTbWaaSbaaWqaaiaadQgaaeqaaSGaaiilai % abgkHiTiaadUgadaWgaaadbaGaamOAaaqabaWccaGGPaaaaOGaaiyk % aaaa!4AA7! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ (hm_n^{({m_j}, - {k_j})}) $$ are biorthogonal systems and that % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaacaWGObWaa0baaSqaaiaad6gaaeaacaGG % OaGaamyBamaaBaaameaacaWGQbaabeaaliaacYcacaWGRbWaaSbaaW % qaaiaadQgaaeqaaSGaaiykaaaaaaa!4765!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ h_n^{({m_j},{k_j})} $$ is the Haar system if m _j = −1, k _j = 0, the Franklin system if m _j =0, k _j =0.