0} s^{1/p} f^\#(s) } & if $q = \infty$ .\cr} $$ Note that $L_{p,q}$\ is not a normed space unless $1 \le q \le p \le \infty$. Note that the $L_p$\ spaces are special cases: $L_p = L_\Phi = L_{p,p}$, where $\Phi(t) = t^p$. We refer the reader to [{\bf 13}] for more details about these spaces. In the present paper we will be interested in the domination of a rearrangement invariant norm of a sum of an arbitrary sequence of adapted random variables by the rearrangement invariant norm of a sum of its decoupled version. It is already known (see [{\bf 4}]) that if $\Phi$\ satisfies the $\Delta_2$-condition, that is, there is a constant $c>0$\ such that $\Phi(2t) \le c\, \Phi(t)$\ for all $t \ge 0$, then there is a constant $C_\Phi$\ such that for every adapted sequence $(f_n)$ of random variables one has: $$ \|\sum f_i\|_\Phi \leq C_\Phi \|\sum\overline f_i\|_\Phi . \eqno(1.1)$$ Building on some special situations considered by Klass [{\bf 4}, Theorem 3.1] and Kwapie\'n [{\bf 9}], Hitczenko [{\bf 6}] began to investigate how the constant $C_\Phi$\ depends upon $\Phi$. He showed that there is a universal constant $C>0$\ such that $$ \|\sum f_i\|_p \leq C\, \|\sum\overline f_i\|_p \qquad 1 \le p \le \infty . \eqno(1.2)$$ In this present paper, we will show among other things, that inequality $(1.1)$\ holds with $C_\Phi$\ uniformly bounded, at least for certain classes of Orlicz functions. Our first theorem extends a result of Klass who proved (1.1) for randomly stopped sums of independent random variables. Let us define classes of Orlicz functions. Following Klass [{\bf 8}], for $q>0$ we define the class $F_q$ as the class of all functions $\Phi$ such that: \item{(i)} $\Phi:[0,\infty)\rightarrow[0,\infty)$, $\Phi(0)=0$, \item{(ii)} $\Phi$ is nondecreasing and continuous, and \item{(iii)} $\Phi$ satisfies the growth condition: $\Phi(cx)\le c^q\Phi(x)$ for all $x\ge 0$, $c\ge2$. \noindent For $p >0$, we define the class $G_p$\ as the class of all functions $\Phi$ such that \item{(i)} $\Phi:[0,\infty)\rightarrow[0,\infty)$, $\Phi(0)=0$, \item{(ii)} $\Phi$ is nondecreasing and continuous, and \item{(iii)} $\Phi$ satisfies the growth condition: $\Phi(cx)\ge c^p\Phi(x)$ for all $x\ge 0$, $c\ge2$. Then we obtain the following results. \proclaim Theorem 1.1. There is a universal constant $C>0$\ such that if $\Phi \in F_q$\ for some $q > 0$, then for every adapted sequence $(f_n)$ of random variables one has: $$ E \Phi\left(\modo{\sum f_i}\right) \leq C^{1+q} E \Phi\left(\modo{\sum \overline f_i}\right).$$ This inequality had already been obtained by Klass [{\bf 8}] in the special case that $f_k=I(\tau\ge k)\xi_k$, where $(\xi_k)$ is a sequence of independent random variables and $\tau$ is a stopping time. More precisely, Klass proved his result for Banach space valued random variables $(\xi_k)$ (with absolute value replaced by norm). To discuss Banach space valued random variables one needs to adjust notation; for a random variable $Y$ and a $\sigma$-algebra ${\cal A}$, we use ${\cal L}(Y |{\cal A})$ to denote the regular version of the conditional distribution of $Y$ given ${\cal A}$, that is, ${\cal L}(Y |{\cal A}) = {\cal L}(Z |{\cal A})$ means that for every Borel subset $A$\ of the Banach space, we have $P(Y\in A|{\cal A}) = P(Z\in A|{\cal A})$. Recall that the existence of the regular versions of the conditional distributions is guaranteed, as long as our random variables take values in a separable Banach space. As it turns out, in our generality, the inequality of Theorem~1.1 need not hold (with {\it any\/} constant), unless some extra conditions are imposed on the geometry of the underlying Banach space (see e.g. [{\bf 3}]). Since it is unclear at this time for which Banach spaces the inequality $$ \big(E\|\sum f_k\|^p\big)^{1/p}\le c_p\big(E\|\sum \ov{f}_k\|^p\big)^{1/p} $$ holds (even if the constant $c_p$ is allowed to depend on $p$), we confine our discussion to real valued random variables. \proclaim Corollary 1.2. Given numbers $p_0>0$\ and $r\ge 1$, there is a constant $c_{p_0,r}$\ such that if $p \ge p_0$, and if $\Phi \in G_p \cap F_{rp}$, then $$ \|\sum f_i\|_\Phi \leq c_{p_0,r} \|\sum\overline f_i\|_\Phi.$$ The next step is to extend these results to rearrangement invariant spaces. This will be accomplished through a rather general method of obtaining rearrangement invariant norm inequalities from Orlicz norm inequalities. We believe that this technique will prove useful in other contexts as well. In particular, we would like to mention that this method could be used to deduce martingale inequalities obtained by Johnson and Schechtman [{\bf 7}] from the corresponding inequalities for Orlicz functions. Corresponding to the notions of Orlicz spaces lying in $G_p \cap F_q$, we have the following notion. We say that a rearrangement invariant space is an {\it interpolation space for $(L_p,L_q)$\/} (in short, a {\it $(p,q)$-interpolation space\/}) if there is a constant $c > 0$ such that for every operator $T:L_p \cap L_q \to L_p \cap L_q$\ for which $\normo T_{L_p \to L_p} \le 1$\ and $\normo T_{L_q \to L_q} \le 1$\ we have that $\normo T_{X \to X} \le c$. However, this notion is not quite what we need. Define $$ K_{p,q}(f,t) = \inf\{ \normo{f'}_p + t \normo{f''}_q : f' + f'' = f^\# \} .$$ We will say that a rearrangement invariant space $X$\ is a {\it $(p,q)$-$K$-interpolation space\/} if there is a constant $c$\ such that whenever $f$\ and $g$\ are such that $K_{p,q}(f,t) \le K_{p,q}(g,t)$\ ($t > 0$), and $g \in X$, then $f \in X$\ and $\normo f_X \le c \normo g_X$. The {\it $(p,q)$-$K$-interpolation constant\/} of $X$, denoted by $C_{p,q}(X)$, is the infimum of $c$\ that work for all functions $f$\ and $g$. It is quite easily seen that every $(p,q)$-$K$-interpolation space is a $(p,q)$-interpolation space. It is also known that if $1 \le p,q \le \infty$, then every normed $(p,q)$-interpolation space is a $(p,q)$-$K$-interpolation space (see [{\bf 1}]). We are able to establish the following method for obtaining rearrangement invariant inequalities from Orlicz inequalities. \proclaim Theorem 1.3. Suppose that $\normo f_\Phi \le \normo g_\Phi$\ for all $\Phi \in F_q \cap G_p$, where $0 < p < q < \infty$. If $X$\ is a $(p,q)$-$K$-interpolation space, then $\normo f_X \le 2^{2+1/p} C_{p,q}(X) \normo g_X$. \proclaim Corollary 1.4. Given numbers $p_0>0$\ and $r\ge 1$, there is a constant $c_{p_0,r}$\ such that if $p \ge p_0$, and if $X$\ is a $(p,pr)$-$K$-interpolation space, then $$ \|\sum f_i\|_X \leq c_{p_0,r} C_{p,pr}(X) \|\sum\overline f_i\|_X.$$ In particular, we are able to obtain the following result for Lorentz spaces. Note that if one is interested in normed Lorentz spaces, then $p_0$ below can be taken to be 1, and the resulting inequality extends (1.2). \proclaim Corollary 1.5. Given a number $p_0>0$, there is a constant $c_{p_0}$\ such that if $p,q \ge p_0$, then $$ \|\sum f_i\|_{p,q} \leq c_{p_0} \|\sum\overline f_i\|_{p,q}.$$ \beginsection 2. Inequalities for Orlicz functions We begin with a proof of Theorem~1.1. Since our proof is based on well understood techniques we will be somewhat sketchy and we refer the reader to [{\bf 6}] for details that are not explained here. Throughout this section we let $(M_n)$ be a martingale with difference sequence $(\Delta_k)$. Since $F_{q_1}\subset F_{q_2}$ whenever $q_1\le q_2$, we can assume without loss of generality that $q\ge 1$. Our departing point is the following result which can be found in the just mentioned paper (Theorem~5.1 and the beginning of the proof of Lemma~2.3). \proclaim Lemma 2.1. Let $1\le q<\infty$, and let $(\Delta_k)$ be a conditionally symmetric martingale difference sequence, and $(\ov{\Delta }_k)$ its decoupled version. Set $T_{n,q}(M)=(E|\sum_{k=1}^n\ov {\Delta}_k|^q\big|{\cal G})^{1/q}$. Then there exist $\delta_1>0$, $\beta>1+\delta_1$ and $\epsilon$\ with $0<\epsilon\le 1/2$ such that for every $\lambda>0$ we have $$ P(M^*\geq \beta\lambda, (T_q^*(M)\vee \Delta^*)<\delta_1\lambda) \leq \epsilon^qP(M^*\ge\lambda). $$ \noindent From this, we obtain \proclaim Lemma 2.2. Let $(\Delta_k)$ be as above, and assume that $w_n$ is a ${\cal F}_{n-1}$-measurable random variable such that $|\Delta_n|\le w_n$\ for each $n \ge 1$. Set $N_n=\sum_{k=1}^n\ov {\Delta}_k$ ($n\ge 1$). Suppose that $\delta_1$, $\beta$, and $\epsilon$ are as in Lemma~2.1. Then, there exist $\delta>0$, $\delta_2>0$ and $0<\alpha\le 1/2$ such that for every $\lambda>0$ we have $$ P(M^*\geq \beta\lambda, N^*<\delta\lambda) \leq \epsilon^qP(M^*\ge\lambda) +P(w^*\ge\delta_2\lambda) + (1-\alpha^q)P(M^*\geq \beta\lambda). $$ \noindent{\bf Proof:} We have that $$ \eqalign{P(M^*\ge\beta\lambda,N^*< \delta\lambda)\le& P(M^*\ge\beta\lambda,T_q^*(M)< \delta_1\lambda, w^*< \delta_2\lambda)+P(w^*\ge\delta_2\lambda)\cr+& P(M^*\ge\beta\lambda,T_q^*(M)\ge \delta_1\lambda,w^*<\delta_2\lambda,N^*< \delta\lambda). \cr}\eqno(2.1) $$ Suppose $\delta_2\le\delta_1$. Then, in view of Lemma 2.1, for the first probability on the right-hand side of (2.1) we have that $$ \eqalign{P(M^*\ge\beta\lambda,T_q^*(M)< \delta_1\lambda, w^*< \delta_2\lambda)\le& P(M^*\ge\beta\lambda,T_q^*(M)\vee w^*< \delta_1\lambda)\cr\le& \epsilon^qP(M^*\ge\lambda).\cr} $$ It remains to estimate the last probability in (2.1). Since $M^*$, $w^*$ and $T_q^*(M)$ are ${\cal G}$-measurable, by conditioning on ${\cal G}$, we see that the last probability in (2.1) is equal to: $$ E\Big\{I(M^*\ge\beta\lambda,T_q^*(M)\ge \delta_1\lambda,w^*<\delta_2\lambda)P(N^*< \delta\lambda|{\cal G})\Big\}. $$ By Kolmogorov's converse inequality (see e.g. [{\bf 12}, Remark 6.15, p. 161]) for all sequences of independent and symmetric random variables $(\xi_k)$ and for all $t>0$ we have that $$ P(S^*\ge t)\ge {1\over 2^q}\Big(1-{2^{2q}(t^q+E(\xi^*)^q)\over E(S^*)^q}\Big), $$ where $S_n=\sum_{k=1}^n\xi_k$. Applying this result conditionally on ${\cal G}$, we obtain that $$ P(N^*<\delta\lambda|{\cal G})\le 1-{1\over 2^q}\Big(1-{2^{2q}((\delta\lambda)^q+E_{\cal G}(\ov {\Delta}^*)^q)\over E_{\cal G}(N^*)^q}\Big).\eqno(2.2) $$ Also, if $w_n<\delta_2\lambda$, then $|\Delta_n|<\delta_2\lambda$, and since $w_n$ is ${\cal F}_{n-1}$-measurable, and the conditional distributions of $\Delta_n$ and $\ov {\Delta}_n$ coincide, it follows that $|\ov{\Delta}_n|< \delta_2\lambda$. Therefore, on the set $$ \{T_q^*(M)\ge\delta_1\lambda,w^*<\delta_2\lambda\}, $$ we have $$ {(\delta\lambda)^q+E_{\cal G}(\ov{ \Delta}^*)^q\over E_{\cal G}(N^*)^q}\le{(\delta\lambda)^q+(\delta_2\lambda)^q\over (\delta_1\lambda)^q}, $$ so that the conditional probability in (2.2) does not exceed $$ 1-{1\over 2^q}\Big(1-{4^q(\delta^q+\delta_2^q)\over \delta_1^q}\Big). $$ Choosing $\delta=\delta_2=\delta_1/12$, we obtain that $$ 1-{1\over 2^q}\Big(1-{4^q(\delta^q+\delta_2^q)\over \delta_1^q}\Big)=1-{1\over 2^q}\Big(1-{2\over 3^q}\Big)\le 1-\alpha^q, $$ whenever $\alpha\le 1/6$. Therefore, $$ \eqalign{ E\Big\{I(M^*\ge\beta\lambda&,T_q(M)\ge \delta_1\lambda,w^*<\delta_2\lambda)P(N^*< \delta\lambda|{\cal G})\Big\}\cr\le& (1-\alpha^q) EI(M^*\ge\beta\lambda,T_q(M)\ge \delta_1\lambda,w^*<\delta_2\lambda) \le(1-\alpha^q)P(M^*\ge\beta\lambda).\cr} $$ This completes the proof of Lemma 2.2. \bigskip \noindent Now we are ready to complete the proof of Theorem~1.1. By an argument similar to one used in [{\bf 6}, proof of Lemma~2.1], it follows that in order to prove $$ E\Phi(|\sum f_k|)\le c^qE\Phi(|\sum\ov{f}_k|),\eqno(2.3) $$ it suffices to establish (2.3) for $(f_k)=(\Delta_k)$, a conditionally symmetric martingale difference sequence. By a routine application of Davis' decomposition (cf. e.g. [{\bf 2}] and references therein), we may also assume that $|\Delta_n|\le w_n$, where $w_n$ is a ${\cal F}_{n-1}$-measurable random variable, and that $w^*\le 2\Delta^*$. The latter inequality, together with the inequality $P(f^*\ge t)\le 2P(g^*\ge t)$ valid for all tangent sequences $(f_k)$ and $(g_k)$ (cf. [{\bf 4}] or [{\bf 11}, Theorem 5.2.1 (i)]), implies that $$ P(w^*\ge t)\le P(\Delta^*\ge t/2)\le 2P(\ov{\Delta}^*\ge t/2)\le 2P(N^*\ge t/4).\eqno(2.4) $$ By Lemma 2.2, we have that $$ P(M^*\ge\beta\lambda)\le P(N^*\ge\delta\lambda)+P(w^*\ge\delta_2\lambda)+ (1-\alpha^q)P(M^*\ge\beta\lambda), $$ so that $$ \alpha^qP(M^*\ge\beta\lambda)\le P(N^*\ge\delta\lambda)+P(w^*\ge\delta_2\lambda). $$ Consequently, $$ \alpha^qE\Phi(M^*/\beta)\le E\Phi(N^*/\delta)+E\Phi(w^*/\delta_2). $$ Therefore, $$ \eqalign{\Big({\alpha\over\beta}\Big)^qE\Phi(M^*)=& \Big({\alpha\over\beta}\Big)^qE\Phi(\beta M^*/\beta)\cr \le&\Big({\alpha\over\beta}\Big)^q\beta^pE\Phi( M^*/\beta)\le E\Phi(N^*/\delta)+E\Phi(w^*/\delta_2)\cr \le&\delta^{-q}E\Phi(N^*)+\delta_2^{-q}E\Phi(w^*).\cr} $$ By (2.4) we have that $$ E\Phi(w^*)\le 2E\Phi(4N^*)\le 2\cdot4^qE\Phi(N^*), $$ so that $$ E\Phi(M^*)\le \Big({\beta\over\alpha}\Big)^q\Big\{{1\over\delta^q} +{2\cdot4^q\over\delta_2^q}\Big\}E\Phi(N^*). $$ Conditionally given ${\cal G}$, $(\ov\Delta_k)$ is a sequence of independent and symmetric random variables. Therefore, by Levy's inequality, $$P(N^*\ge t\big|{\cal G})\le 2P(N\ge t\big|{\cal G}).$$ This implies (see e.g. [{\bf 11}, Proposition 0.2.1]) that for every increasing function $\phi:{\bf R}^+\to{\bf R}^+$\ we have $E\phi(N^*)\le2E\phi(N)$. This completes the proof of Theorem 1.1. As we mentioned in the introduction, inequality (2.3) extends a result of Klass, who considered sequences $(f_k)$ of special form. On the other hand, it follows from a result of Kwapie\'n [{\bf 9}] that if $f_k=(\sum_{j=1}^{k-1}a_{j,k}\xi_j)\xi_k$, where $(\xi_j)$ is a sequence of independent zero mean random variables, then (2.3) holds in the stronger form: $$ E\Phi(|\sum f_k|)\le E\Phi(c|\sum\ov f_k|), $$ for some absolute constant c and for {\it every} convex function $\Phi$. Thus one may wonder whether our restrictions on $\Phi$ can be relaxed. We wish to close this section with a negative result showing that (2.3) does not hold for all convex functions $\Phi$. (However, it is still possible that (2.3) holds under weaker assumption than ours.) Our example is an easy adaptation of an example due to Talagrand concerning comparison of tail behavior for sums of tangent sequences. This example was included in [{\bf 5}], and we refer the reader to the latter paper for details that are not included here. \proclaim Proposition 2.3. For every constant $c>0$, there exists a convex function $\Phi:\bf R\to\bf R$ and a sequence $(f_k)$ such that $$ E\Phi(|\sum f_k|)\ge cE\Phi(c|\sum\overline{f}_k|). $$ \noindent{\bf Proof:}\ \ We will show that for every $k\in{\bf N}$ there exists a convex function $\Phi$ and a sequence $(f_k)$ for which $$ E\Phi(|\sum f_k|)\ge {2^{2^{k+2}}\over k^22^{2k}}E\Phi({k\over4}|\sum\overline{f}_k|). $$ Let $(r_n)$\ denote the Rademacher random variables, that is, a sequence of independent random variables such that $P(r_n = \pm 1) = 1/2$. Fix $k\in{\bf N}$. Given an integer $N_1$ to be specified in a moment, define $N_2,\dots N_k$ as follows: $$ N_i-N_{i-1}=2^{-(i-1)}N_1,\quad i=2,\dots ,k. $$ Put $$ \Omega _1=\{ r_1=\dots =r_{N_1}=1\} , $$ and then $$ \Omega _i=\Omega _{i-1}\cap\{ r_{N_{i-1}+1}=\dots =r_{N_i}\},\quad i=2,\dots ,k. $$ Define a sequence of random variables $(v_i)$ by the formulas: $$ \eqalign {v_1=\dots =v_{N_1}&=1\cr v_{N_1+1}=\dots =v_{N_2}&=2I_{\Omega _1}\cr \dots\dots\dots\dots &\cr v_{N_{k-1}+1}=\dots =v_{N_k}&=2^{k-1}I_{\Omega _{k-1}}.\cr} $$ We let $f_j=v_jr_j$ for $j=1,\dots N_k$. Then $\overline{f}_j=v_jr_j'$, where $(r_j')$ is an independent copy of $(r_j)$ (cf. [{\bf 10}, Example 4.3.1]). For $0<\delta<1$, let $\Phi_{\delta}$ be a convex function defined by $\Ph(x)=(x-\delta kN_1)^+$. Note that $\sum |v_j|=kN_1$, and therefore $$ E\Ph(|\sum v_jr_j|)\ge (1-\d)kN_1P(|\sum v_jr_j|\ge kN_1). $$ On the other hand, if $|\sum v_jr_j'|<4N_1$, then $|\sum v_jr_j'|\le 4N_1-1$, so that with $\d=1-1/(4N_1)$ we get $$ \Ph({k\over4}|\sum v_jr_j'|)\le(kN_1-{k\over 4}-\d kN_1)^+=0. $$ Since $$ P(|\sum v_jr_j'|\ge 4N_1)\le k2^{-N_1/2^{k-2}}P(|\sum v_jr_j|\ge kN_1), $$ (cf. [{\bf 5}, top half of page 176]) we obtain $$ \eqalign{E\Ph({k\over4}|\sum v_jr_j'|)\le& \Ph({k^2N_1\over4})P(|\sum v_jr_j'|\ge 4N_1)\cr \le& kN_1{k\over4}k2^{-N_1/2^{k-2}}P(|\sum v_jr_j|\ge kN_1),\cr} $$ and it follows that $$ {E\Ph(|\sum v_jr_j|)\over E\Ph((k/4)|\sum v_jr_j'|)}\ge {4(1-\d)kN_12^{N_1/2^{k-2}}\over k^3N_1}= {2^{N_1/2^{k-2}} \over k^2N_1}\ge{2^{2^{k+2}}\over k^22^{2k}}, $$ for $N_1\ge 2^{k}$. This completes the proof. In the above example the sequence $(f_k)$ may be constructed so that $\sum f_k$ is a randomly stopped sum of independent random variables (see Remark on p. 176 of [{\bf 5}]). Thus, the conclusion of this Remark applies here as well. \beginsection 3. Rearrangement invariant norm inequalities \proclaim Lemma 3.1. Suppose that $\Phi$\ and $\Psi$\ are two Orlicz functions. Let $\Theta = \Phi \wedge \Psi$, and $\Theta_1(x) = {1\over 2} \Theta(x)$. Then $$ \textstyle{1\over 2}\normo f_{\Theta_1} \le \inf\{ \normo{f'}_\Phi + \normo{f''}_\Psi : f' + f'' = f^\#\} \le 2\,\normo f_{\Theta} .$$ \noindent{\bf Proof:} To show the left hand side, suppose that $f^\# = f' + f''$, and $\normo{f'}_\Phi + \normo{f''}_\Psi \le 1$. Then $E\Phi(\modo{f'}) \le 1$\ and $E\Psi(\modo{f''}) \le 1$, and so $$ \eqalign{ E\Theta_1(\textstyle{1\over 2}|f|) \le& \textstyle{1\over 2} E\Theta(\max\{\modo{f'},\modo{f''}\}) = \textstyle{1\over 2} E \max\{\Theta(\modo{f'}),\Theta(\modo{f''})\} \cr \le& \textstyle {1\over 2} (E\Phi(|f'|) + E\Psi(|f''|)) \le 1 .\cr}$$ To show the right hand side, suppose that $\normo f_{\Theta} \le 1$, that is, $E\Theta(|f|) \le 1$. Let $$ f'(t) = \cases{ f^\#(t) & if $\Phi(|f(t)|) \le \Psi(|f(t)|)$ \cr 0 & otherwise,\cr} $$ and $f'' = f^\# - f'$. Then we see that $E\Phi(|f'|) \le E\Theta(\modo{f}) \le 1$\ and that $E\Psi(|f''|) \le E\Theta(\modo{f}) \le 1$, and the result follows. The next lemma follows immediately. \proclaim Lemma 3.2. Let $\Phi_t(x) = x^p \wedge (tx)^q$, where $0 < p < q < \infty$. Then $$ 2^{-1-1/p} \normo f_{\Phi_t} \le K_{p,q}(f,t) \le 2 \normo f_{\Phi_t} .$$ Now we will prove Theorem~1.3, using the above Lemma. From the hypothesis of Theorem~3.1, it follows that $\normo f_{\Phi_t} \le \normo g_{\Phi_t}$. Hence by Lemma 3.2, it follows that $K_{p,q}(f,t) \le 2^{2+1/p}K_{p,q}(g,t) $. Now the result follows by the definition of $(p,q)$-$K$-interpolation space. Corollary~1.4 follows easily from Theorems 1.1 and 1.3. To show Corollary~1.5, we only need the following result. The methods below are all fairly standard in interpolation theory, and indeed if one is not concerned about uniform estimates, may be taken directly from the literature. \proclaim Lemma 3.3. Given $p_0>0$, there is a constant $c_{p_0}>0$\ such that if $p,q \ge p_0$, then $L_{p,q}$\ is a $(p/2,2p)$-$K$-interpolation space with constant bounded by $c_{p_0}$. \noindent {\bf Proof:}\ \ First let us define some norms. For $p\le q$, let $$ \normo f_{a(t)} = \inf\{ t^{-2/p} \normo{f'}_{p/2} + t^{-1/2p} \normo{f''}_{2p} : f'+f'' = f^\# \} ,$$ $$ \normo f_{b(t)} = \left( {1\over t} \int_0^t f^\#(s)^{p/2} \, ds \right)^{2/p} + \left( {1\over t} \int_t^\infty f^\#(s)^{2p} \, ds \right)^{1/2p} .$$ Clearly $\normo f_{a(t)} \le \normo f_{b(t)}$. Also $\normo f_{a(t)} \ge \min\{1,2^{1-2/p}\} f^\#(t)$. This is because if $f^\# = f' + f''$, then $$ \eqalignno{ t^{-2/p} \normo{f'}_{p/2} + t^{-1/2p} \normo{f''}_{2p} &\ge \left({1\over t} \int_0^t \modo{f'(s)}^{p/2} \, ds \right)^{2/p} + \left({1\over t} \int_0^t \modo{f''(s)}^{2p} \, ds \right)^{1/2p} \cr &\ge \left({1\over t} \int_0^t \modo{f'(s)}^{p/2} \, ds \right)^{2/p} + \left({1\over t} \int_0^t \modo{f''(s)}^{p/2} \, ds \right)^{2/p} \cr &\ge \min\{1,2^{1-2/p}\} \left({1\over t} \int_0^t f^\#(s)^{p/2} \, ds \right)^{2/p} \cr &\ge \min\{1,2^{1-2/p}\} f^\#(t) .\cr }$$ Next, given a function $f$, let us define the function $Hf(t) = \normo f_{b(t)}$. Then it follows that $\normo{Hf}_{p,q} \le 32^{1/\min\{p,q\}}\normo f_{p,q}$. To see this, first note that $\normo{Hf}_{p,q} \le (\normo{H_1f}_{p,q}^q + \normo{H_2f}_{p,q}^q)^{1/q}$, where $$ \eqalignno{ H_1 f(t) &= \left({1\over t} \int_0^t f^\#(s)^{p/2} \, ds, \right)^{2/p} \cr H_2 f(t) &= \left({1\over t} \int_t^\infty f^\#(s)^{2p} \, ds \right)^{1/2p} \cr} $$ We will use two properties of $L_{p,q}$. First, if $v \le q$, and if $f_1$, $f_2,\dots,$\ $f_n$\ are functions, then $$ \normo{\left(\sum_{i=1}^n (f_i^\#)^v\right)^{1/v} }_{p,q} \le \left(\sum_{i=1}^n \normo {f_i}_{p,q}^v \right)^{1/v} .$$ Second, if we define the operators $D_af(t) = f^\#(at)$\ for $0 < a < \infty$, then $\normo{D_af}_{p,q} = a^{-1/p} \normo f_{p,q}$. (The first property follows from Minkowski's inequality for $L_ {q/v}$, the second is a simple change of variables argument.) Let $u = \max\{p/q,2\}$, $v = \min\{q,p/2\}$. Then $$ \eqalignno{ \normo{H_1f}_{p,q} &\le \normo{\left(\int_0^1 (D_a f^\#)^{p/2} \, da \right)^{2/p}}_{p,q} \cr &\le \normo{\left(\sum_{n=0}^\infty \int_{2^{-u (n+1)}}^{2^{-u n}} (D_a f^\#)^{p/2} \, da \right)^{2/p} }_{p,q} \cr &\le \normo{\left(\sum_{n=0}^\infty 2^{-u n} (D_{2^{-u (n+1)}} f^\#)^{p/2} \right)^{2/p} }_{p,q} \cr &\le \normo{\left(\sum_{n=0}^\infty 4^{-n} (D_{2^{-u(n+1)}} f^\#)^v \right)^{1/v} }_{p,q} \cr &\le \left( \sum_{n=0}^\infty 4^{-n} \normo{D_{2^{-u(n+1)}} f^\#}_{p,q}^v \right)^{1/v} \cr &\le \left( \sum_{n=0}^\infty 4^{-n} 2^{u(n+1)v/p})\right)^{1/v} \normo f_{p,q} \cr &\le \left( \sum_{n=0}^\infty 2^{1-n} \right)^{1/v} \normo f_{p,q} \cr &\le 4^{1/v} \normo f_{p,q} .\cr}$$ Now let $u = \max\{2p/q,1\}$, $v = \min\{q,2p\}$. $$ \eqalignno{ \normo{H_2f}_{p,q} &\le \normo{\left(\int_1^\infty (D_a f^\#)^{2p} \, da \right)^{1/2p}}_{p,q} \cr &\le \normo{\left(\sum_{n=0}^\infty \int_{2^{u n}}^{2^{u(n+1)}} (D_a f^\#)^{2p} \, da \right)^{1/2p} }_{p,q} \cr &\le \normo{\left(\sum_{n=0}^\infty 2^{u(n+1)} (D_{2^{u n}} f^\#)^{2p} \right)^{1/2p} }_{p,q} \cr &\le \normo{\left(\sum_{n=0}^\infty 2^{n+1} (D_{2^{u n}} f^\#)^v \right)^{1/v} }_{p,q} \cr &\le \left( \sum_{n=0}^\infty 2^{n+1} \normo{D_{2^{u n}} f^\#}_{p,q}^v \right)^{1/v} \cr &\le \left( \sum_{n=0}^\infty 2^{n+1} 2^{-u n v/p})\right)^{1/v} \normo f_{p,q} \cr &\le \left( \sum_{n=0}^\infty 2^{1-n} \right)^{1/v} \normo f_{p,q} \cr &\le 4^{1/v} \normo f_{p,q} .\cr}$$ Finally, to finish, suppose that $K_{p/2,2p}(f,t) \le K_{p/2,2p}(g,t)$\ for all $t > 0$. Then it follows that $$ \eqalign{f^\#(t) \le& \max\{1,2^{2/p-1}\} \normo f_{a(t)} \le \max\{1,2^{2/p-1}\} \normo g_{a(t)} \le \max\{1,2^{2/p-1}\} \normo g_{b(t)} \cr =& 2^{2/p} Hg(t) .\cr} $$ Hence $$ \normo f_{p,q} \le 2^{2/p} \normo{Hg}_{p,q} \le 128^{1/\min\{p,q\}} \normo g_{p,q} .$$ \beginsection Acknowledgments The second named author would like to express warm gratitude to Victor de la Pe\~na for introducing him to this problem. Research of both authors was partially supported by separate NSF grants. The second named author was also supported by the Research Board of the University of Missouri. \beginsection References \medskip \frenchspacing \newcount\refnum \refnum=0 \def\ref{\global\advance\refnum by 1\item{[{\bf\the\refnum}]}} \ref J. ARAZY and M. CWIKIEL. A new characterization of the interpolation spaces between $L^p$\ and $L^q$, {\it Math. Scand.} {\bf 55} (1984), 253--270. \ref D. L. BURKHOLDER. Distribution function inequalities for martingales, {\it Ann. Prob\-ab.} {\bf 1} (1973), 19 - 42. \ref D. J. H. GARLING. Random martingale transform inequalities, {\it Probability in Banach Spaces, 6 (Sandbjerg, Denmark, 1986), 101 -- 119, Progr. Probab.} {\bf 20}, Birkh\"auser, Boston (1990). \ref P. HITCZENKO. Comparison of moments for tangent sequences of random variables. {\it Probab. Theory Related Fields} {\bf 78} (1988), 223 -- 230. \ref P. HITCZENKO. Domination inequality for martingale transforms of a Radema\-cher sequence. {\it Israel J. Math.} {\bf 84} (1993), 161 -- 178. \ref P. HITCZENKO. On a domination of sums of random variables by sums of conditionally independent ones. {\it Ann. Probab.} {\bf 22} (1994), 453 -- 468. \ref W. B. JOHNSON and G. SCHECHTMAN. Martingale inequalities in rearrangement invariant function spaces, {\it Israel J. Math.} {\bf 64} (1988), 267 - 275. \ref M. J. KLASS. A best possible improvement of Wald's equation, {\it Ann. Probab.} {\bf 16} (1988), 840 - 853. \ref S. KWAPIE\'N. Decoupling inequalities for polynomial chaos, {\it Ann. Probab.} {\bf 15} (1987), 1062 - 1072. \ref S. KWAPIE\'N and W. A. WOYCZY\'NSKI. Semimartingale integrals via decoupling inequalities and tangent processes, {\it Probab. Math. Statist.} {\bf 12} (1991), 165 -- 200. \ref S. KWAPIE\'N and W. A. WOYCZY\'NSKI. {\it Random Series and Stochastic Integrals. Single and Multiple}, Birkh\"auser, Boston (1992). \ref M. LEDOUX and M. TALAGRAND. {\it Probability in Banach Spaces.} Springer, Berlin, Heidelberg (1991). \ref J. LINDENSTRAUSS and L. TZAFRIRI. {\it Classical Banach Spaces. Function Spaces}, Springer, Berlin, Heidelberg, New York (1977). \ref S. J. MONTGOMERY -- SMITH. Comparison of Orlicz -- Lorentz spaces, {\it Studia Math.} {\bf 103} (1992), 161 -- 189. \bye